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# RD Sharma solutions for Class 10 Mathematics chapter 6 - Co-Ordinate Geometry

## Chapter 6: Co-Ordinate Geometry

Ex. 6.10Ex. 6.20Ex. 6.30Ex. 8.00Ex. 6.40Ex. 6.50Others

#### Chapter 6: Co-Ordinate Geometry Exercise 6.10 solutions [Page 4]

Ex. 6.10 | Q 1.1 | Page 4

On which axis do the following points lie?

P(5, 0)

Ex. 6.10 | Q 1.1 | Page 4

On which axis do the following points lie?

P(5, 0)

Ex. 6.10 | Q 1.2 | Page 4

On which axis do the following points lie?

Q(0, -2)

Ex. 6.10 | Q 1.2 | Page 4

On which axis do the following points lie?

Q(0, -2)

Ex. 6.10 | Q 1.3 | Page 4

On which axis do the following points lie?

R(−4,0)

Ex. 6.10 | Q 1.3 | Page 4

On which axis do the following points lie?

R(−4,0)

Ex. 6.10 | Q 1.4 | Page 4

On which axis do the following points lie?

S(0,5)

Ex. 6.10 | Q 1.4 | Page 4

On which axis do the following points lie?

S(0,5)

Ex. 6.10 | Q 2.1 | Page 4

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when A coincides with the origin and AB and AD are along OX and OY respectively.

Ex. 6.10 | Q 2.1 | Page 4

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when A coincides with the origin and AB and AD are along OX and OY respectively.

Ex. 6.10 | Q 2.2 | Page 4

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when The centre of the square is at the origin and coordinate axes are parallel to the sides AB and AD respectively.

Ex. 6.10 | Q 2.2 | Page 4

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when The centre of the square is at the origin and coordinate axes are parallel to the sides AB and AD respectively.

Ex. 6.10 | Q 3 | Page 4

The base PQ of two equilateral triangles PQR and PQR' with side 2a lies along y-axis such that the mid-point of PQ is at the origin. Find the coordinates of the vertices R and R' of the triangles.

Ex. 6.10 | Q 3 | Page 4

The base PQ of two equilateral triangles PQR and PQR' with side 2a lies along y-axis such that the mid-point of PQ is at the origin. Find the coordinates of the vertices R and R' of the triangles.

#### Chapter 6: Co-Ordinate Geometry Exercise 6.20 solutions [Pages 15 - 17]

Ex. 6.20 | Q 1.1 | Page 15

Find the distance between the following pair of points:

(-6, 7) and (-1, -5)

Ex. 6.20 | Q 1.1 | Page 15

Find the distance between the following pair of points:

(-6, 7) and (-1, -5)

Ex. 6.20 | Q 1.2 | Page 15

Find the distance between the following pair of points:

(a+b, b+c) and (a-b, c-b)

Ex. 6.20 | Q 1.2 | Page 15

Find the distance between the following pair of points:

(a+b, b+c) and (a-b, c-b)

Ex. 6.20 | Q 1.3 | Page 15

Find the distance between the following pair of points:

(asinα, −bcosα) and (−acos α, bsin α)

Ex. 6.20 | Q 1.3 | Page 15

Find the distance between the following pair of points:

(asinα, −bcosα) and (−acos α, bsin α)

Ex. 6.20 | Q 1.4 | Page 15

Find the distance between the following pair of points:

(a, 0) and (0, b)

Ex. 6.20 | Q 1.4 | Page 15

Find the distance between the following pair of points:

(a, 0) and (0, b)

Ex. 6.20 | Q 2 | Page 15

Find the value of a when the distance between the points (3, a) and (4, 1) is sqrt10

Ex. 6.20 | Q 2 | Page 15

Find the value of a when the distance between the points (3, a) and (4, 1) is sqrt10

Ex. 6.20 | Q 3 | Page 15

If the points (2, 1) and (1, -2) are equidistant from the point (xy), show that x + 3y = 0.

Ex. 6.20 | Q 3 | Page 15

If the points (2, 1) and (1, -2) are equidistant from the point (xy), show that x + 3y = 0.

Ex. 6.20 | Q 4 | Page 15

Find the values of x, y if the distances of the point (xy) from (-3, 0)  as well as from (3, 0) are 4.

Ex. 6.20 | Q 4 | Page 15

Find the values of x, y if the distances of the point (xy) from (-3, 0)  as well as from (3, 0) are 4.

Ex. 6.20 | Q 5 | Page 15

The length of a line segment is of 10 units and the coordinates of one end-point are (2, -3). If the abscissa of the other end is 10, find the ordinate of the other end.

Ex. 6.20 | Q 5 | Page 15

The length of a line segment is of 10 units and the coordinates of one end-point are (2, -3). If the abscissa of the other end is 10, find the ordinate of the other end.

Ex. 6.20 | Q 6 | Page 15

Show that the points (−4, −1), (−2, −4) (4, 0) and (2, 3) are the vertices points of a rectangle.

Ex. 6.20 | Q 6 | Page 15

Show that the points (−4, −1), (−2, −4) (4, 0) and (2, 3) are the vertices points of a rectangle.

Ex. 6.20 | Q 7 | Page 15

Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.

Ex. 6.20 | Q 7 | Page 15

Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.

Ex. 6.20 | Q 8 | Page 15

Prove that the points A(1, 7), B (4, 2), C(−1, −1) D (−4, 4) are the vertices of a square.

Ex. 6.20 | Q 8 | Page 15

Prove that the points A(1, 7), B (4, 2), C(−1, −1) D (−4, 4) are the vertices of a square.

Ex. 6.20 | Q 9 | Page 15

Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right-angled isosceles triangle.

Ex. 6.20 | Q 9 | Page 15

Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right-angled isosceles triangle.

Ex. 6.20 | Q 10 | Page 15

Prove that (2, -2) (-2, 1) and (5, 2) are the vertices of a right-angled triangle. Find the area of the triangle and the length of the hypotenuse.

Ex. 6.20 | Q 11 | Page 15

Prove that the points (2a, 4a), (2a, 6a) and (2a + sqrt3a, 5a) are the vertices of an equilateral triangle.

Ex. 6.20 | Q 12 | Page 15

Prove that the points (2,3), (-4, -6) and (1, 3/2) do not form a triangle.

Ex. 6.20 | Q 14 | Page 15

Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.

Ex. 6.20 | Q 14 | Page 15

Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.

Ex. 6.20 | Q 15 | Page 15

Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.

Ex. 6.20 | Q 15 | Page 15

Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.

Ex. 6.20 | Q 16 | Page 16

Which point on the x-axis is equidistant from (5, 9) and (−4, 6)?

Ex. 6.20 | Q 16 | Page 16

Which point on the x-axis is equidistant from (5, 9) and (−4, 6)?

Ex. 6.20 | Q 17 | Page 16

Prove that the points (−2, 5), (0, 1) and (2, −3)  are collinear.

Ex. 6.20 | Q 17 | Page 16

Prove that the points (−2, 5), (0, 1) and (2, −3)  are collinear.

Ex. 6.20 | Q 18 | Page 16

The coordinates of the point P are (−3, 2). Find the coordinates of the point Q which lies on the line joining P and origin such that OP = OQ.

Ex. 6.20 | Q 18 | Page 16

The coordinates of the point P are (−3, 2). Find the coordinates of the point Q which lies on the line joining P and origin such that OP = OQ.

Ex. 6.20 | Q 19 | Page 16

Which point on the y-axis is equidistant from (2, 3)  and (−4, 1)?

Ex. 6.20 | Q 19 | Page 16

Which point on the y-axis is equidistant from (2, 3)  and (−4, 1)?

Ex. 6.20 | Q 20 | Page 16

The three vertices of a parallelogram are (3, 4) (3, 8) and (9, 8). Find the fourth vertex.

Ex. 6.20 | Q 20 | Page 16

The three vertices of a parallelogram are (3, 4) (3, 8) and (9, 8). Find the fourth vertex.

Ex. 6.20 | Q 24 | Page 16

Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5).

Ex. 6.20 | Q 24 | Page 16

Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5).

Ex. 6.20 | Q 26 | Page 16

Show that the points (−3, 2), (−5,−5), (2, −3) and (4, 4) are the vertices of a rhombus. Find the area of this rhombus.

Ex. 6.20 | Q 26 | Page 16

Show that the points (−3, 2), (−5,−5), (2, −3) and (4, 4) are the vertices of a rhombus. Find the area of this rhombus.

Ex. 6.20 | Q 27 | Page 16

Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (-1, -6) and (4, -1). Also, find its circumradius.

Ex. 6.20 | Q 27 | Page 16

Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (-1, -6) and (4, -1). Also, find its circumradius.

Ex. 6.20 | Q 28 | Page 16

Find a point on the x-axis which is equidistant from the points (7, 6) and (−3, 4).

Ex. 6.20 | Q 28 | Page 16

Find a point on the x-axis which is equidistant from the points (7, 6) and (−3, 4).

Ex. 6.20 | Q 29.1 | Page 16

Show that the points A(5, 6), B(1, 5), C(2, 1) and D(6,2) are the vertices of a square.

Ex. 6.20 | Q 29.1 | Page 16

Show that the points A(5, 6), B(1, 5), C(2, 1) and D(6,2) are the vertices of a square.

Ex. 6.20 | Q 29.2 | Page 16

Prove hat the points A (2, 3) B(−2,2) C(−1,−2), and D(3, −1) are the vertices of a square ABCD.

Ex. 6.20 | Q 30 | Page 16

Find the point on x-axis which is equidistant from the points (−2, 5) and (2,−3).

Ex. 6.20 | Q 30 | Page 16

Find the point on x-axis which is equidistant from the points (−2, 5) and (2,−3).

Ex. 6.20 | Q 31 | Page 16

Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, -1), (1, 3) and (x, 8) respectively.

Ex. 6.20 | Q 31 | Page 16

Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, -1), (1, 3) and (x, 8) respectively.

Ex. 6.20 | Q 32 | Page 16

Prove that the points (0, 0), (5, 5) and (-5, 5) are the vertices of a right isosceles triangle.

Ex. 6.20 | Q 32 | Page 16

Prove that the points (0, 0), (5, 5) and (-5, 5) are the vertices of a right isosceles triangle.

Ex. 6.20 | Q 33 | Page 16

If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.

Ex. 6.20 | Q 33 | Page 16

If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.

Ex. 6.20 | Q 34 | Page 16

If Q (0, 1) is equidistant from P (5, -3) and R (x, 6), find the values of x. Also, find the
distances QR and PR

Ex. 6.20 | Q 34 | Page 16

If Q (0, 1) is equidistant from P (5, -3) and R (x, 6), find the values of x. Also, find the
distances QR and PR

Ex. 6.20 | Q 35 | Page 16

Find the values of y for which the distance between the points P (2, -3) and Q (10, y) is
10 units

Ex. 6.20 | Q 35 | Page 16

Find the values of y for which the distance between the points P (2, -3) and Q (10, y) is
10 units

Ex. 6.20 | Q 36 | Page 16

If the point P(k-1, 2) is equidistant from the points A(3,k) and B(k,5), find the value of k.

Ex. 6.20 | Q 37 | Page 17

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.

Ex. 6.20 | Q 37 | Page 17

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.

Ex. 6.20 | Q 38.1 | Page 17

A(-1,-2) B(1, 0), C (-1, 2), D(-3, 0)

Ex. 6.20 | Q 38.1 | Page 17

A(-1,-2) B(1, 0), C (-1, 2), D(-3, 0)

Ex. 6.20 | Q 38.2 | Page 17

A(-3, 5) B(3, 1), C (0, 3), D(-1, -4)

Ex. 6.20 | Q 38.2 | Page 17

A(-3, 5) B(3, 1), C (0, 3), D(-1, -4)

Ex. 6.20 | Q 38.3 | Page 17

A(4, 5) B(7, 6), C (4, 3), D(1, 2)

Ex. 6.20 | Q 38.3 | Page 17

A(4, 5) B(7, 6), C (4, 3), D(1, 2)

Ex. 6.20 | Q 39 | Page 17

Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3,5).

Ex. 6.20 | Q 39 | Page 17

Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3,5).

Ex. 6.20 | Q 40 | Page 17

Prove that the points (3, 0), (4, 5), (-1, 4) and (-2, -1), taken in order, form a rhombus.
Also, find its area.

Ex. 6.20 | Q 40 | Page 17

Prove that the points (3, 0), (4, 5), (-1, 4) and (-2, -1), taken in order, form a rhombus.
Also, find its area.

Ex. 6.20 | Q 41 | Page 17

In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at A(3, 1), B(6, 4), and C(8, 6). Do you think they are seated in a line?

Ex. 6.20 | Q 41 | Page 17

In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at A(3, 1), B(6, 4), and C(8, 6). Do you think they are seated in a line?

Ex. 6.20 | Q 42 | Page 17

Find a point on y-axis which is equidistant from the points (5, -2) and (-3, 2).

Ex. 6.20 | Q 42 | Page 17

Find a point on y-axis which is equidistant from the points (5, -2) and (-3, 2).

Ex. 6.20 | Q 43 | Page 17

Find a relation between x and y such that the point (xy) is equidistant from the points (3, 6) and (-3, 4).

Ex. 6.20 | Q 43 | Page 17

Find a relation between x and y such that the point (xy) is equidistant from the points (3, 6) and (-3, 4).

Ex. 6.20 | Q 44 | Page 17

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.

Ex. 6.20 | Q 44 | Page 17

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.

Ex. 6.20 | Q 45 | Page 17

prove  that the points A (7, 10), B(-2, 5) and C(3, -4) are the vertices of an isosceles right triangle.

Ex. 6.20 | Q 46 | Page 17

If the point P(x, 3) is equidistant from the point A(7, −1) and B(6, 8), then find the value of x and find the distance AP.

Ex. 6.20 | Q 46 | Page 17

If the point P(x, 3) is equidistant from the point A(7, −1) and B(6, 8), then find the value of x and find the distance AP.

Ex. 6.20 | Q 47 | Page 17

If A(3, y) is equidistant from points P(8, −3) and Q(7, 6), find the value of y and find the distance AQ.

Ex. 6.20 | Q 47 | Page 17

If A(3, y) is equidistant from points P(8, −3) and Q(7, 6), find the value of y and find the distance AQ.

Ex. 6.20 | Q 48 | Page 17

If (0, −3) and (0, 3) are the two vertices of an equilateral triangle, find the coordinates of its third vertex.

Ex. 6.20 | Q 48 | Page 17

If (0, −3) and (0, 3) are the two vertices of an equilateral triangle, find the coordinates of its third vertex.

Ex. 6.20 | Q 49 | Page 17

If the point P(2, 2) is equidistant from the points A(−2, k) and B(−2k, −3), find k. Also find the length of AP.

Ex. 6.20 | Q 49 | Page 17

If the point P(2, 2) is equidistant from the points A(−2, k) and B(−2k, −3), find k. Also find the length of AP.

Ex. 6.20 | Q 50 | Page 17

Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles.

Ex. 6.20 | Q 50 | Page 17

Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles.

Ex. 6.20 | Q 51 | Page 17

An equilateral triangle has two vertices at the points (3, 4) and (−2, 3), find the coordinates of the third vertex.

Ex. 6.20 | Q 51 | Page 17

An equilateral triangle has two vertices at the points (3, 4) and (−2, 3), find the coordinates of the third vertex.

Ex. 6.20 | Q 52 | Page 17

Find the circumcenter of the triangle whose vertices are (-2, -3), (-1, 0), (7, -6).

Ex. 6.20 | Q 52 | Page 17

Find the circumcenter of the triangle whose vertices are (-2, -3), (-1, 0), (7, -6).

Ex. 6.20 | Q 53 | Page 17

Find the angle subtended at the origin by the line segment whose end points are (0, 100) and (10, 0).

Ex. 6.20 | Q 54 | Page 17

Find the centre of the circle passing through (5, -8), (2, -9) and (2, 1).

Ex. 6.20 | Q 54 | Page 17

Find the centre of the circle passing through (5, -8), (2, -9) and (2, 1).

Ex. 6.20 | Q 55 | Page 17

If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.

Ex. 6.20 | Q 55 | Page 17

If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.

Ex. 6.20 | Q 56 | Page 17

Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)

Ex. 6.20 | Q 56 | Page 17

Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)

Ex. 6.20 | Q 57 | Page 17

Two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of other two
vertices.

Ex. 6.20 | Q 57 | Page 17

Two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of other two
vertices.

#### Chapter 6: Co-Ordinate Geometry Exercise 6.30, 8.00 solutions [Pages 28 - 31]

Ex. 6.30 | Q 1 | Page 28

Find the coordinates of the point which divides the line segment joining (−1,3) and (4, −7) internally in the ratio 3 : 4

Ex. 6.30 | Q 1 | Page 28

Find the coordinates of the point which divides the line segment joining (−1,3) and (4, −7) internally in the ratio 3 : 4

Ex. 6.30 | Q 2.1 | Page 28

Find the points of trisection of the line segment joining the points:

5, −6 and (−7, 5),

Ex. 6.30 | Q 2.1 | Page 28

Find the points of trisection of the line segment joining the points:

5, −6 and (−7, 5),

Ex. 6.30 | Q 2.2 | Page 28

Find the points of trisection of the line segment joining the points:

(3, -2) and (-3, -4)

Ex. 6.30 | Q 2.2 | Page 28

Find the points of trisection of the line segment joining the points:

(3, -2) and (-3, -4)

Ex. 6.30 | Q 2.3 | Page 28

Find the points of trisection of the line segment joining the points:

(2, -2) and (-7, 4).

Ex. 6.30 | Q 2.3 | Page 28

Find the points of trisection of the line segment joining the points:

(2, -2) and (-7, 4).

Ex. 6.30 | Q 3 | Page 28

Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (-2, -1), (1, 0), (4, 3) and(1, 2) meet

Ex. 6.30 | Q 3 | Page 28

Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (-2, -1), (1, 0), (4, 3) and(1, 2) meet

Ex. 6.30 | Q 4 | Page 28

Prove that the points (3, -2), (4, 0), (6, -3) and (5, -5) are the vertices of a parallelogram.

Ex. 6.30 | Q 4 | Page 28

Prove that the points (3, -2), (4, 0), (6, -3) and (5, -5) are the vertices of a parallelogram.

Ex. 6.30 | Q 5 | Page 28

If P ( 9a -2  , - b) divides the line segment joining A (3a + 1 , - 3 ) and B (8a, 5) in the ratio 3 : 1 , find the values of a and b .

Ex. 6.30 | Q 5 | Page 28

If P ( 9a -2  , - b) divides the line segment joining A (3a + 1 , - 3 ) and B (8a, 5) in the ratio 3 : 1 , find the values of a and b .

Ex. 6.30 | Q 6 | Page 28

If (a,b) is the mid-point of the line segment joining the points A (10, - 6) , B (k,4) and a - 2b = 18 , find the value of k and the distance AB.

Ex. 6.30 | Q 7 | Page 29

Find the ratio in which the point (2, y) divides the line segment joining the points A (-2,2) and B (3, 7). Also, find the value of y.

Ex. 6.30 | Q 7 | Page 29

Find the ratio in which the point (2, y) divides the line segment joining the points A (-2,2) and B (3, 7). Also, find the value of y.

Ex. 8.00 | Q 8 | Page 29

If A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.

Ex. 8.00 | Q 8 | Page 29

If A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.

Ex. 6.30 | Q 9 | Page 29

If the points P, Q(x, 7), R, S(6, y) in this order divide the line segment joining A(2, p) and B(7, 10) in 5 equal parts, find xy and p

Ex. 6.30 | Q 10 | Page 29

If a vertex of a triangle be (1, 1) and the middle points of the sides through it be (-2,-3) and (5 2) find the other vertices.

Ex. 6.30 | Q 10 | Page 29

If a vertex of a triangle be (1, 1) and the middle points of the sides through it be (-2,-3) and (5 2) find the other vertices.

Ex. 6.30 | Q 11.1 | Page 29

In what ratio is the line segment joining the points (-2,-3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

Ex. 6.30 | Q 11.1 | Page 29

In what ratio is the line segment joining the points (-2,-3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

Ex. 6.30 | Q 11.2 | Page 29

In what ratio is the line segment joining (-3, -1) and (-8, -9) divided at the point (-5, -21/5)?

Ex. 6.30 | Q 11.2 | Page 29

In what ratio is the line segment joining (-3, -1) and (-8, -9) divided at the point (-5, -21/5)?

Ex. 6.30 | Q 12 | Page 29

If the mid-point of the line joining (3, 4) and (k, 7) is (x, y) and 2x + 2y + 1 = 0 find the value of k.

Ex. 6.30 | Q 12 | Page 29

If the mid-point of the line joining (3, 4) and (k, 7) is (x, y) and 2x + 2y + 1 = 0 find the value of k.

Ex. 6.30 | Q 13 | Page 29

Find the ratio in which the point  $\left( \frac{3}{4}, \frac{5}{12} \right)$  divides the line segment joining the points A $\left( \frac{1}{2}, \frac{3}{2} \right)$  and B $\left( 2, - 5 \right)$ .

Ex. 6.30 | Q 14.1 | Page 29

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by x-axis Also, find the coordinates of the point of division in each case.

Ex. 6.30 | Q 14.1 | Page 29

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by x-axis Also, find the coordinates of the point of division in each case.

Ex. 6.30 | Q 14.2 | Page 29

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by y-axis. Also, find the coordinates of the point of division in each case.

Ex. 6.30 | Q 14.2 | Page 29

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by y-axis. Also, find the coordinates of the point of division in each case.

Ex. 6.30 | Q 15 | Page 29

Prove that the points (4, 5) (7, 6), (6, 3) (3, 2) are the vertices of a parallelogram. Is it a rectangle.

Ex. 6.30 | Q 15 | Page 29

Prove that the points (4, 5) (7, 6), (6, 3) (3, 2) are the vertices of a parallelogram. Is it a rectangle.

Ex. 6.30 | Q 16 | Page 29

Prove that (4, 3), (6, 4) (5, 6) and (3, 5)  are the angular points of a square.

Ex. 6.30 | Q 16 | Page 29

Prove that (4, 3), (6, 4) (5, 6) and (3, 5)  are the angular points of a square.

Ex. 6.30 | Q 17 | Page 29

Prove that the points A(-4,-1), B(-2, 4), C(4, 0) and D(2, 3) are the vertices of a rectangle.

Ex. 6.30 | Q 17 | Page 29

Prove that the points A(-4,-1), B(-2, 4), C(4, 0) and D(2, 3) are the vertices of a rectangle.

Ex. 6.30 | Q 18 | Page 29

Find the lengths of the medians of a triangle whose vertices are A (−1,3), B(1,−1) and C(5, 1).

Ex. 6.30 | Q 18 | Page 29

Find the lengths of the medians of a triangle whose vertices are A (−1,3), B(1,−1) and C(5, 1).

Ex. 6.30 | Q 19 | Page 29

Find the ratio in which the line segment joining the points A(3, −3) and B(−2, 7) is divided by the x-axis. Also, find the coordinates of the point of division.

Ex. 6.30 | Q 20 | Page 29

Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, −3). Also, find the value of x.

Ex. 6.30 | Q 20 | Page 29

Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, −3). Also, find the value of x.

Ex. 6.30 | Q 21 | Page 29

Find the ratio in which the point (-1, y) lying on the line segment joining points A(-3, 10) and (6, -8) divides it. Also, find the value of y.

Ex. 6.30 | Q 22 | Page 29

Find the coordinates of a point A, where AB is a diameter of the circle whose centre is (2, -3) and B is (1, 4).

Ex. 6.30 | Q 22 | Page 29

Find the coordinates of a point A, where AB is a diameter of the circle whose centre is (2, -3) and B is (1, 4).

Ex. 6.30 | Q 23 | Page 29

If the points (-2, -1), (1, 0), (x, 3) and  (1, y) form a parallelogram, find the values of x and y.

Ex. 6.30 | Q 23 | Page 29

If the points (-2, -1), (1, 0), (x, 3) and  (1, y) form a parallelogram, find the values of x and y.

Ex. 6.30 | Q 24 | Page 29

The points A(2, 0), B(9, 1) C(11, 6) and D(4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.

Ex. 6.30 | Q 24 | Page 29

The points A(2, 0), B(9, 1) C(11, 6) and D(4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.

Ex. 6.30 | Q 25 | Page 29

In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?

Ex. 6.30 | Q 25 | Page 29

In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?

Ex. 6.30 | Q 26 | Page 29

Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–1, –4). Also find the coordinates of the point of division.

Ex. 6.30 | Q 26 | Page 29

Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–1, –4). Also find the coordinates of the point of division.

Ex. 6.30 | Q 27 | Page 29

Show that A (−3, 2), B (−5, −5), (2,−3), and D (4, 4) are the vertices of a rhombus.

Ex. 6.30 | Q 27 | Page 29

Show that A (−3, 2), B (−5, −5), (2,−3), and D (4, 4) are the vertices of a rhombus.

Ex. 6.30 | Q 28 | Page 29

Find the length of the medians of a ΔABC having vertices at A(0, -1), B(2, 1) and C(0, 3).

Ex. 6.30 | Q 28 | Page 29

Find the length of the medians of a ΔABC having vertices at A(0, -1), B(2, 1) and C(0, 3).

Ex. 6.30 | Q 29 | Page 29

Find the lengths of the medians of a ΔABC having vertices at A(5, 1), B(1, 5), and C(-3, -1).

Ex. 6.30 | Q 29 | Page 29

Find the lengths of the medians of a ΔABC having vertices at A(5, 1), B(1, 5), and C(-3, -1).

Ex. 6.30 | Q 30 | Page 29

Find the coordinates of the points which divide the line segment joining the points (-4, 0) and (0, 6) in four equal parts.

Ex. 6.30 | Q 30 | Page 29

Find the coordinates of the points which divide the line segment joining the points (-4, 0) and (0, 6) in four equal parts.

Ex. 6.30 | Q 31 | Page 29

Show that the mid-point of the line segment joining the points (5, 7) and (3, 9) is also the mid-point of the line segment joining the points (8, 6) and (0, 10).

Ex. 6.30 | Q 31 | Page 29

Show that the mid-point of the line segment joining the points (5, 7) and (3, 9) is also the mid-point of the line segment joining the points (8, 6) and (0, 10).

Ex. 6.30 | Q 32 | Page 29

Find the distance of the point (1, 2) from the mid-point of the line segment joining the points (6, 8) and (2, 4).

Ex. 6.30 | Q 32 | Page 29

Find the distance of the point (1, 2) from the mid-point of the line segment joining the points (6, 8) and (2, 4).

Ex. 6.30 | Q 33 | Page 29

If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.

Ex. 6.30 | Q 33 | Page 29

If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.

Ex. 6.30 | Q 34 | Page 29

Show that the points A (1, 0), B (5, 3), C (2, 7) and D (−2, 4) are the vertices of a parallelogram.

Ex. 6.30 | Q 34 | Page 29

Show that the points A (1, 0), B (5, 3), C (2, 7) and D (−2, 4) are the vertices of a parallelogram.

Ex. 6.30 | Q 35 | Page 29

Determine the ratio in which the point P (m, 6) divides the join of A(-4, 3) and B(2, 8). Also, find the value of m.

Ex. 6.30 | Q 35 | Page 29

Determine the ratio in which the point P (m, 6) divides the join of A(-4, 3) and B(2, 8). Also, find the value of m.

Ex. 6.30 | Q 36 | Page 29

Determine the ratio in which the point (-6, a) divides the join of A (-3, 1)  and B (-8, 9). Also, find the value of a.

Ex. 6.30 | Q 36 | Page 29

Determine the ratio in which the point (-6, a) divides the join of A (-3, 1)  and B (-8, 9). Also, find the value of a.

Ex. 6.30 | Q 37 | Page 30

ABCD is a rectangle formed by the points A(-1, -1), B(-1, 4), C(5, 4) and D(5, -1). P, Q, R and S are the midpoints of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

Ex. 6.30 | Q 38 | Page 30

Points P, Q, R and S divides the line segment joining A(1, 2) and B(6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R.

Ex. 6.30 | Q 39 | Page 30

If A and B are (− 2, − 2) and (2, − 4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.

Ex. 6.30 | Q 39 | Page 30

If A and B are (− 2, − 2) and (2, − 4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.

Ex. 6.30 | Q 40 | Page 30

Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.

Ex. 6.30 | Q 40 | Page 30

Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.

Ex. 6.30 | Q 41 | Page 30

Three consecutive vertices of a parallelogram are (-2,-1), (1, 0) and (4, 3). Find the fourth vertex.

Ex. 6.30 | Q 41 | Page 30

Three consecutive vertices of a parallelogram are (-2,-1), (1, 0) and (4, 3). Find the fourth vertex.

Ex. 6.30 | Q 42 | Page 30

The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the coordinates of the fourth vertex.

Ex. 6.30 | Q 42 | Page 30

The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the coordinates of the fourth vertex.

Ex. 6.30 | Q 43 | Page 30

If the coordinates of the mid-points of the sides of a triangle are (1, 1), (2, —3) and (3, 4), find the vertices of the triangle.

Ex. 6.30 | Q 44 | Page 30

Determine the ratio in which the straight line x - y - 2 = 0 divides the line segment
joining (3, -1) and (8, 9).

Ex. 6.30 | Q 44 | Page 30

Determine the ratio in which the straight line x - y - 2 = 0 divides the line segment
joining (3, -1) and (8, 9).

Ex. 6.30 | Q 45 | Page 30

Three vertices of a parallelogram are (a+b, a-b), (2a+b, 2a-b), (a-b, a+b). Find the fourth vertex.

Ex. 6.30 | Q 45 | Page 30

Three vertices of a parallelogram are (a+b, a-b), (2a+b, 2a-b), (a-b, a+b). Find the fourth vertex.

Ex. 6.30 | Q 46 | Page 30

If two vertices of a parallelogram are (3, 2) (-1, 0) and the diagonals cut at (2, -5), find the other vertices of the parallelogram.

Ex. 6.30 | Q 46 | Page 30

If two vertices of a parallelogram are (3, 2) (-1, 0) and the diagonals cut at (2, -5), find the other vertices of the parallelogram.

Ex. 6.30 | Q 47 | Page 30

If the coordinates of the mid-points of the sides of a triangle are (3, 4) (4, 6) and (5, 7), find its vertices.

Ex. 6.30 | Q 48 | Page 30

The line segment joining the points P(3, 3) and Q(6, -6) is trisected at the points A and B such that Ais nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.

Ex. 6.30 | Q 48 | Page 30

The line segment joining the points P(3, 3) and Q(6, -6) is trisected at the points A and B such that Ais nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.

Ex. 6.30 | Q 49 | Page 30

If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its fourth vertex.

Ex. 6.30 | Q 49 | Page 30

If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its fourth vertex.

Ex. 6.30 | Q 50 | Page 30

If the points A (a, -11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.

Ex. 6.30 | Q 50 | Page 30

If the points A (a, -11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.

Ex. 6.30 | Q 51 | Page 31

If the coordinates of the mid-points of the sides of a triangle be (3, -2), (-3, 1) and (4, -3), then find the coordinates of its vertices.

Ex. 6.30 | Q 51 | Page 31

If the coordinates of the mid-points of the sides of a triangle be (3, -2), (-3, 1) and (4, -3), then find the coordinates of its vertices.

Ex. 6.30 | Q 52 | Page 31

The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, -2) and (5/3, q) respectively. Find the values of p and q.

Ex. 6.30 | Q 52 | Page 31

The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, -2) and (5/3, q) respectively. Find the values of p and q.

Ex. 6.30 | Q 53 | Page 31

The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If point P lies on the line 2x - y + k = 0. Find the value of k.

Ex. 6.30 | Q 53 | Page 31

The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If point P lies on the line 2x - y + k = 0. Find the value of k.

Ex. 6.30 | Q 54 | Page 31

Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ΔABC.

(i) The median from A meets BC at D. Find the coordinates of point D.

(ii) Find the coordinates of the point P on AD such that AP: PD = 2:1

(iii) Find the coordinates of point Q and R on medians BE and CF respectively such that BQ: QE = 2:1 and CR: RF = 2:1.

(iv) What do you observe?

(v) If A(x1y1), B(x2y2), and C(x3y3) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

Ex. 6.30 | Q 55 | Page 31

If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.

Ex. 6.30 | Q 55 | Page 31

If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.

Ex. 6.30 | Q 56 | Page 31

A point P divides the line segment joining the points A(3, -5) and B(-4, 8) such that (AP)/(PB) = k/1. If P lies on the line x + y = 0, then find the value of k.

Ex. 6.30 | Q 56 | Page 31

A point P divides the line segment joining the points A(3, -5) and B(-4, 8) such that (AP)/(PB) = k/1. If P lies on the line x + y = 0, then find the value of k.

Ex. 6.30 | Q 57 | Page 31

The midpoint P of the line segment joining points A(-10, 4) and B(-2, 0) lies on the line segment joining the points C(-9, -4) and D(-4, y). Find the ratio in which P divides CD. Also, find the value of y.

Ex. 6.30 | Q 58 | Page 31

If the point  $C \left( - 1, 2 \right)$ divides internally the line segment joining the points  A (2, 5)  and Bx) in the ratio 3 : 4 , find the value of x2 + y2 .

Ex. 6.30 | Q 59 | Page 31

ABCD is a parallelogram with vertices  $A ( x_1 , y_1 ), B \left( x_2 , y_2 \right), C ( x_3 , y_3 )$   . Find the coordinates  of the fourth vertex D in terms of  $x_1 , x_2 , x_3 , y_1 , y_2 \text{ and } y_3$

Ex. 6.30 | Q 60 | Page 31

The points  $A \left( x_1 , y_1 \right) , B\left( x_2 , y_2 \right) , C\left( x_3 , y_3 \right)$   are the vertices of  ΔABC .
(i) The median from meets BC at D . Find the coordinates of the point  D.
(ii) Find the coordinates of the point on AD such that AP : PD  = 2 : 1.
(iii) Find the points of coordinates Q and on medians BE and CF respectively such thatBQ : QE = 2 : 1 and CR : RF = 2 : 1.
(iv) What are the coordinates of the centropid of the triangle ABC

#### Chapter 6: Co-Ordinate Geometry Exercise 6.40 solutions [Page 37]

Ex. 6.40 | Q 1.1 | Page 37

Find the centroid of the triangle whosw vertices is  (1,4), (-1,1) and (3,2) .

Ex. 6.40 | Q 1.2 | Page 37

Find the centroid of the triangle whose vertices  is (−2, 3) (2, −1) (4, 0) .

Ex. 6.40 | Q 2 | Page 37

Two vertices of a triangle are (1, 2), (3, 5) and its centroid is at the origin. Find the coordinates of the third vertex.

Ex. 6.40 | Q 3 | Page 37

Find the third vertex of a triangle, if two of its vertices are at (−3, 1) and (0, −2) and the centroid is at the origin.

Ex. 6.40 | Q 3 | Page 37

Find the third vertex of a triangle, if two of its vertices are at (−3, 1) and (0, −2) and the centroid is at the origin.

Ex. 6.40 | Q 4 | Page 37

A (3, 2) and B (−2, 1)  are two vertices of a triangle ABC whose centroid G has the coordinates (5/3,-1/3)Find the coordinates of the third vertex C of the triangle.

Ex. 6.40 | Q 4 | Page 37

A (3, 2) and B (−2, 1)  are two vertices of a triangle ABC whose centroid G has the coordinates (5/3,-1/3)Find the coordinates of the third vertex C of the triangle.

Ex. 6.40 | Q 5 | Page 37

If (−2, 3), (4, −3) and (4, 5) are the mid-points of the sides of a triangle, find the coordinates of its centroid.

Ex. 6.40 | Q 5 | Page 37

If (−2, 3), (4, −3) and (4, 5) are the mid-points of the sides of a triangle, find the coordinates of its centroid.

Ex. 6.40 | Q 6 | Page 37

Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.

Ex. 6.40 | Q 7 | Page 37

Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another

Ex. 6.40 | Q 8 | Page 37

If G be the centroid of a triangle ABC and P be any other point in the plane, prove that PA2+ PB2 + PC2 = GA2 + GB2 + GC2 + 3GP2.

Ex. 6.40 | Q 9 | Page 37

If G be the centroid of a triangle ABC, prove that:

AB2 + BC2 + CA2 = 3 (GA2 + GB2 + GC2)

Ex. 6.40 | Q 9 | Page 37

If G be the centroid of a triangle ABC, prove that:

AB2 + BC2 + CA2 = 3 (GA2 + GB2 + GC2)

Ex. 6.40 | Q 10 | Page 37

In Fig. 14.36, a right triangle BOA is given C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices O, A  and B.

We have a right angled triangle,triangle BOA  right angled at O. Co-ordinates are B (0,2b); A (2a0) and C (0, 0).

Ex. 6.40 | Q 10 | Page 37

In Fig. 14.36, a right triangle BOA is given C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices O, A  and B.

We have a right angled triangle,triangle BOA  right angled at O. Co-ordinates are B (0,2b); A (2a0) and C (0, 0).

#### Chapter 6: Co-Ordinate Geometry Exercise 6.50 solutions [Pages 53 - 55]

Ex. 6.50 | Q 1.1 | Page 53

Find the area of a triangle whose vertices are

(6,3), (-3,5) and (4,2)

Ex. 6.50 | Q 1.2 | Page 53

Find the area of a triangle whose vertices are

(at_1^2,2at_1),(at_2^2,2at_2) and (at_3^2,2at_3)

Ex. 6.50 | Q 1.3 | Page 53

Find the area of a triangle whose vertices are

(a, c + a), (a, c) and (−a, c − a)

Ex. 6.50 | Q 2.1 | Page 53

Find the area of the quadrilaterals, the coordinates of whose vertices are

(−3, 2), (5, 4), (7, −6) and (−5, −4)

Ex. 6.50 | Q 2.2 | Page 53

Find the area of the quadrilaterals, the coordinates of whose vertices are

(1, 2), (6, 2), (5, 3) and (3, 4)

Ex. 6.50 | Q 2.3 | Page 53

Find the area of the quadrilateral whose vertices, taken in order, are (-4, -2), (-3, -5), (3, -2) and (2, 3).

Ex. 6.50 | Q 3 | Page 53

The four vertices of a quadrilateral are (1, 2), (−5, 6), (7, −4) and (k, −2) taken in order. If the area of the quadrilateral is zero, find the value of k.

Ex. 6.50 | Q 4 | Page 53

The vertices of ΔABC are (−2, 1), (5, 4)  and (2, −3)  respectively. Find the area of the triangle and the length of the altitude through A.

Ex. 6.50 | Q 5.1 | Page 53

Show that the following sets of points are collinear.

(2, 5), (4, 6) and (8, 8)

Ex. 6.50 | Q 5.2 | Page 53

Show that the following sets of points are collinear.

(1, −1), (2, 1) and (4, 5)

Ex. 6.50 | Q 7 | Page 54

In  $∆$ ABC , the coordinates of vertex A are (0, - 1) and D (1,0) and E(0,10)  respectively the mid-points of the sides AB and AC . If F is the mid-points of the side BC , find the area of $∆$ DEF.

Ex. 6.50 | Q 8 | Page 54

Find the area of the triangle PQR with Q(3,2) and the mid-points of the sides through Q being (2,−1) and (1,2).

Ex. 6.50 | Q 9 | Page 54

If P(–5, –3), Q(–4, –6), R(2, –3) and S(1, 2) are the vertices of a quadrilateral PQRS, find its area.

Ex. 6.50 | Q 10 | Page 54

If A(−3, 5), B(−2, −7), C(1, −8) and D(6, 3) are the vertices of a quadrilateral ABCD, find its area.

Ex. 6.50 | Q 11 | Page 54

For what value of a point (a, 1), (1, -1) and (11, 4) are collinear?

Ex. 6.50 | Q 12 | Page 54

Prove that the points (a, b), (a1, b1) and (a −a1, b −b1) are collinear if ab1 = a1b.

Ex. 6.50 | Q 13 | Page 54

If the vertices of a triangle are (1, −3), (4, p) and (−9, 7) and its area is 15 sq. units, find the value(s) of p.

Ex. 6.50 | Q 14 | Page 54

If (x, y) be on the line joining the two points (1, −3) and (−4, 2) , prove that x + y + 2= 0.

Ex. 6.50 | Q 15 | Page 54

Find the value of k if points A(k, 3), B(6, −2) and C(−3, 4) are collinear.

Ex. 6.50 | Q 16 | Page 54

Find the value of k, if the points A(7, −2), B (5, 1) and (3, 2k) are collinear.

Ex. 6.50 | Q 17 | Page 54

If the point P (m, 3) lies on the line segment joining the points $A\left( - \frac{2}{5}, 6 \right)$ and B (2, 8), find the value of m.

Ex. 6.50 | Q 18 | Page 54

If R (x, y) is a point on the line segment joining the points P (a, b) and Q (b, a), then prove that y = a + b.

Ex. 6.50 | Q 19 | Page 54

Find the value of k, if the points A (8, 1) B(3, −4) and C(2, k) are collinear.

Ex. 6.50 | Q 20 | Page 54

Find the value of a for which the area of the triangle formed by the points A(a, 2a), B(−2, 6) and C(3, 1) is 10 square units.

Ex. 6.50 | Q 21 | Page 54

If a≠b≠0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.

Ex. 6.50 | Q 21 | Page 54

If a≠b≠0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.

Ex. 6.50 | Q 22 | Page 54

The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is (7/2, y). Find the value of y

Ex. 6.50 | Q 23 | Page 54

Prove that the points (a, 0), (0, b) and (1, 1) are collinear if 1/a+1/b=1

Ex. 6.50 | Q 24 | Page 54

The point A divides the join of P (−5, 1)  and Q(3, 5) in the ratio k:1. Find the two values of k for which the area of ΔABC where B is (1, 5) and C(7, −2) is equal to 2 units.

Ex. 6.50 | Q 25 | Page 54

The area of a triangle is 5. Two of its vertices are (2, 1) and (3, −2). The third vertex lies on y = x + 3. Find the third vertex.

Ex. 6.50 | Q 26 | Page 54

If a≠ b ≠ c, prove that the points (a, a2), (bb2), (cc2) can never be collinear.

Ex. 6.50 | Q 27 | Page 54

Four points A (6, 3), B (−3, 5), C(4, −2) and D (x, 3x) are given in such a way that (ΔDBG) /(ΔABG)=1/2, find x

Ex. 6.50 | Q 28 | Page 55

If three points (x1, y1) (x2, y2), (x3, y3) lie on the same line, prove that  $\frac{y_2 - y_3}{x_2 x_3} + \frac{y_3 - y_1}{x_3 x_1} + \frac{y_1 - y_2}{x_1 x_2} = 0$

Ex. 6.50 | Q 29 | Page 55

Find the area of a parallelogram ABCD if three of its vertices are A(2, 4), B(2 + $\sqrt{3}$ , 5) and C(2, 6).

Ex. 6.50 | Q 30 | Page 55

Find the value(s) of k for which the points (3k − 1, k − 2), (kk − 7) and (k − 1, −k − 2) are collinear.

Ex. 6.50 | Q 31 | Page 55

If the points A(−1, −4), B(bc) and C(5, −1) are collinear and 2b + c = 4, find the values of b and c.

Ex. 6.50 | Q 32 | Page 55

If the points A(−2, 1), B(a, b) and C(4, −1) ae collinear and a − b = 1, find the values of aand b.

Ex. 6.50 | Q 33 | Page 55

If the points  $A(1, - 2) , B (2, 3) , C (a, 2) \text{ and } D ( - 4, - 3)$ form a parallelogram , find the value of  a  and height of the parallelogram taking  AB as base .

Ex. 6.50 | Q 34 | Page 55

$A\left( 6, 1 \right) , B(8, 2) \text{ and } C(9, 4)$ are three vertices of a parallelogram ABCD . If E is the mid-point  of DC , find the area of  $∆$ ADE.

Ex. 6.50 | Q 35 | Page 55

If  $D\left( - \frac{1}{5}, \frac{5}{2} \right), E(7, 3) \text{ and } F\left( \frac{7}{2}, \frac{7}{2} \right)$  are the mid-points of sides of  $∆ ABC$ ,  find the area of  $∆ ABC$ .

#### Chapter 6: Co-Ordinate Geometry solutions [Pages 61 - 62]

Q 1 | Page 61

Write the distance between the points A (10 cos θ, 0) and B (0, 10 sin θ).

Q 2 | Page 61

Write the perimeter of the triangle formed  by the points O (0, 0), A (a, 0) and B (0, b).

Q 3 | Page 61

Write the ratio in which the line segment joining points (2, 3) and (3, −2) is divided by X axis.

Q 4 | Page 61

What is the distance between the points (5 sin 60°, 0) and (0, 5 sin 30°)?

Q 4 | Page 61

What is the distance between the points (5 sin 60°, 0) and (0, 5 sin 30°)?

Q 5 | Page 61

If A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.

Q 5 | Page 61

If A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.

Q 6 | Page 61

If the distance between points (x, 0) and (0, 3) is 5, what are the values of x?

Q 7 | Page 61

What is the area of the triangle formed by the points O (0, 0), A (6, 0) and B (0, 4)?

Q 7 | Page 61

What is the area of the triangle formed by the points O (0, 0), A (6, 0) and B (0, 4)?

Q 8 | Page 61

Write the coordinates of the point dividing line segment joining points (2, 3) and (3, 4) internally in the ratio 1 : 5.

Q 9 | Page 62

If the centroid of the triangle formed by points P (a, b), Q(b, c) and R (c, a) is at the origin, what is the value of a + b + c?

Q 10 | Page 62

what is the value of  $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}$ .

Q 11 | Page 62

Write the coordinates of a point on X-axis which is equidistant from the points (−3, 4) and (2, 5).

Q 12 | Page 62

If the mid-point of the segment joining A (xy + 1) and B (x + 1, y + 2) is C $\left( \frac{3}{2}, \frac{5}{2} \right)$ , find xy.

Q 13 | Page 62

Two vertices of a triangle have coordinates (−8, 7) and (9, 4) . If the centroid of the triangle is at the origin, what are the coordinates of the third vertex?

Q 14 | Page 62

Write the coordinates the reflections of points (3, 5) in X and Y -axes.

Q 15 | Page 62

If points Q and reflections of point P (−3, 4) in X and Y axes respectively, what is QR?

Q 16 | Page 62

Write the formula for the area of the triangle having its vertices at (x1, y1), (x2, y2) and (x3, y3).

Q 17 | Page 62

Write the condition of collinearity of points (x1, y1), (x2, y2) and (x3, y3).

Q 18 | Page 62

Find the values of x for which the distance between the point P(2, −3), and Q (x, 5) is 10.

Q 19 | Page 62

Write the ratio in which the line segment doining the points A (3, −6), and B (5, 3) is divided by X-axis.

Q 20 | Page 62

Find the distance between the points $\left( - \frac{8}{5}, 2 \right)$  and $\left( \frac{2}{5}, 2 \right)$ .

Q 21 | Page 62

Find the value of a so that the point (3, a) lies on the line represented by 2x − 3y + 5 = 0

Q 22 | Page 62

What is the distance between the points A (c, 0) and B (0, −c)?

Q 23 | Page 62

If P (2, 6) is the mid-point of the line segment joining A (6, 5) and B (4, y), find y.

Q 24 | Page 62

If the distance between the points (3, 0) and (0, y) is 5 units and y is positive. then what is the value of y?

Q 25 | Page 62

If P (x, 6) is the mid-point of the line segment joining A (6, 5) and B (4, y), find y.

Q 26 | Page 62

If P (2, p) is the mid-point of the line segment joining the points A (6, −5) and B (−2, 11). find the value of p.

Q 27 | Page 62

If A (1, 2) B (4, 3) and C (6, 6) are the three vertices of a parallelogram ABCD, find the coordinates of fourth vertex D.

Q 28 | Page 62

What is the distance between the points  $A\left( \sin\theta - \cos\theta, 0 \right)$ and $B\left( 0, \sin\theta + \cos\theta \right)$ ?

Q 30 | Page 62

Find the area of triangle with vertices ( ab+c) , (bc+a) and (ca+b).

Q 31 | Page 62

If the points A (1,2) , O (0,0) and C (a,b) are collinear , then find  a : b.

Q 32 | Page 62

Find the coordinates of the point which is equidistant from the three vertices A ($2x, 0) O (0, 0) \text{ and } B(0, 2y) of ∆$  AOB .

Q 33 | Page 62

If the distance between the points (4, k) and (1, 0) is 5, then what can be the possible values of k?

Q 33 | Page 62

If the distance between the points (4, k) and (1, 0) is 5, then what can be the possible values of k?

#### Chapter 6: Co-Ordinate Geometry solutions [Pages 63 - 67]

Q 1 | Page 63

The distance between the points (cos θ, 0) and (sin θ − cos θ) is

• $\sqrt{3}$

• $\sqrt{2}$

• 2

• 1

Q 2 | Page 63

The distance between the points (a cos 25°, 0) and (0, a cos 65°) is

• a

• 2a

• 3a

•  None of these

Q 3 | Page 63

If x is a positive integer such that the distance between points P (x, 2) and Q (3, −6) is 10 units, then x =

• 3

• -3

• 9

• -9

Q 4 | Page 63

The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ − b cos θ) is

•  a2 + b2

•  a + b

•  a2 − b2

• $\sqrt{a2 + b2}$

Q 5 | Page 63

If the distance between the points (4, p) and (1, 0) is 5, then p =

• ± 4

•  4

•  −4

•  0

Q 6 | Page 63

A line segment is of length 10 units. If the coordinates of its one end are (2, −3) and the abscissa of the other end is 10, then its ordinate is

• 9, 6

•  3, −9

•  −3, 9

•  9, −6

Q 7 | Page 63

The perimeter of the triangle formed by the points (0, 0), (0, 1) and (0, 1) is

•  1 ± $\sqrt{2}$

• $\sqrt{2}$  + 1

• 3

• $2 + \sqrt{2}$

Q 8 | Page 63

If A (2, 2), B (−4, −4) and C (5, −8) are the vertices of a triangle, than the length of the median through vertex C is

• $\sqrt{65}$

• $\sqrt{117}$

• $\sqrt{85}$

• $\sqrt{113}$

Q 9 | Page 63

If three points (0, 0), $\left( 3, \sqrt{3} \right)$  and (3, λ) form an equilateral triangle, then λ =

• 2

• -3

• -4

•  None of these

Q 10 | Page 63

If the points (k, 2k), (3k, 3k) and (3, 1) are collinear, then k

• $\frac{1}{3}$

• $- \frac{1}{3}$

• $\frac{2}{3}$

• $- \frac{2}{3}$

Q 11 | Page 64

The coordinates of the point on X-axis which are equidistant from the points (−3, 4) and (2, 5) are

• (20, 0)

• (−23, 0)

• $\left( \frac{4}{5}, 0 \right)$

• None of these

Q 12 | Page 64

If (−1, 2), (2, −1) and (3, 1) are any three vertices of a parallelogram, then

• a = 2, b = 0

• a = −2, b = 0

• a = −2, = 6

• a = 6, b = 2

• None of these

Q 13 | Page 64

If A (5, 3), B (11, −5) and P (12, y) are the vertices of a right triangle right angled at P, then y=

• −2, 4

•  −2, −4

• 2, −4

•  2, 4

Q 14 | Page 64

The area of the triangle formed by (ab + c), (bc + a) and (ca + b)

•  a + b + c

• abc

• (a + b + c)2

• 0

Q 15 | Page 64

If (x , 2), (−3, −4) and (7, −5) are collinear, then x =

•  60

• 63

• −63

• −60

Q 16 | Page 64

If points (t, 2t), (−2, 6) and (3, 1) are collinear, then t =

• $\frac{3}{4}$

• $\frac{4}{3}$

• $\frac{5}{3}$

• $\frac{3}{5}$

Q 17 | Page 64

If the area of the triangle formed by the points (x, 2x), (−2, 6)  and (3, 1) is 5 square units , then x =

• $\frac{2}{3}$

• $\frac{3}{5}$

• 3

• 5

Q 18 | Page 64

If points (a, 0), (0, b) and (1, 1)  are collinear, then $\frac{1}{a} + \frac{1}{b} =$

• 1

• 2

• 0

• -1

Q 19 | Page 64

If the centroid of a triangle is (1, 4) and two of its vertices are (4, −3) and (−9, 7), then the area of the triangle is

• 183 sq. units

• $\frac{183}{2}$  sq. units

• 366 sq. units

• $\frac{183}{4}$  sq. units

Q 20 | Page 64

The line segment joining points (−3, −4), and (1, −2) is divided by y-axis in the ratio

• 1 : 3

•  2 : 3

• 3 : 1

• 2 : 3

Q 21 | Page 64

The ratio in which (4, 5) divides the join of (2, 3) and (7, 8) is

• −2 : 3

•  −3 : 2

•  3 : 2

• 2 : 3

Q 22 | Page 64

The ratio in which the x-axis divides the segment joining (3, 6) and (12, −3) is

• 2: 1

• 1 : 2

• −2 : 1

•  1 : −2

Q 23 | Page 64

If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, then a3 b3 + c3 =

• abc

• 0

• a + b + c

•  3 abc

Q 24 | Page 64

If Points (1, 2) (−5, 6) and (a, −2) are collinear, then a =

• −3

• 7

• 2

• -2

Q 25 | Page 64

If the centroid of the triangle formed by (7, x) (y, −6) and (9, 10) is at (6, 3), then (x, y) =

• (4, 5)

•  (5, 4)

• (−5, −2)

• (5, 2)

Q 26 | Page 65

The distance of the point (4, 7) from the x-axis is

• 4

• 7

• 11

• $\sqrt{65}$

Q 27 | Page 65

The distance of the point (4, 7) from the y-axis is

• 4

• 7

• 11

• $\sqrt{65}$

Q 28 | Page 65

If P is a point on x-axis such that its distance from the origin is 3 units, then the coordinates of a point on OY such that OP = OQ, are

• (0, 3)

• (3, 0)

• (0, 0)

•  (0, −3)

Q 29 | Page 65

If the points(x, 4) lies on a circle whose centre is at the origin and radius is 5, then x =

•  ±5

•  ±3

• 0

•  ±4

Q 30 | Page 65

If the points P (xy) is equidistant from A (5, 1) and B (−1, 5), then

•  5x = y

• x = 5y

• 3x = 2y

• 2x = 3y

Q 31 | Page 65

If points A (5, pB (1, 5), C (2, 1) and D (6, 2) form a square ABCD, then p =

• 7

• 3

• 6

• 8

Q 32 | Page 65

The coordinates of the circumcentre of the triangle formed by the points O (0, 0), A (a, 0 and B (0, b) are

•  (ab)

• $\left( \frac{a}{2}, \frac{b}{2} \right)$

• $\left( \frac{b}{2}, \frac{a}{2} \right)$

• (ba)

Q 33 | Page 65

The coordinates of a point on x-axis which lies on the perpendicular bisector of the line segment joining the points (7, 6) and (−3, 4) are

• (0, 2)

•  (3, 0)

•  (0, 3)

•  (2, 0)

Q 34 | Page 65

If the centroid of the triangle formed by the points (3, −5), (−7, 4), (10, −k) is at the point (k −1), then k =

• 3

• 1

• 2

• 4

Q 35 | Page 65

If (−2, 1) is the centroid of the triangle having its vertices at (x , 0) (5, −2),  (−8, y), then xy satisfy the relation

• 3x + 8y = 0

•  3x − 8y = 0

• 8x + 3y = 0

• 8x = 3y

• None of these

Q 36 | Page 65

The coordinates of the fourth vertex of the rectangle formed by the points (0, 0), (2, 0), (0, 3) are

• (3, 0)

• (0, 2)

•  (2, 3)

• (3, 2)

Q 37 | Page 65

The length of a line segment joining A (2, −3) and B is 10 units. If the abscissa of B is 10 units, then its ordinates can be

• 3 or −9

• −3 or 9

• 6 or 27

• −6 or −27

Q 38 | Page 65

The ratio in which the line segment joining P (x1y1) and Q (x2, y2) is divided by x-axis is

•  y1 : y2

• −y1 : y2

•  x1 : x2

•  −x1 : x2

Q 39 | Page 65

The ratio in which the line segment joining points A (a1b1) and B (a2b2) is divided by y-axis is

• a1 : a2

•  a1 a2

• b1 : b2

•  −b1 : b2

Q 40 | Page 66

If the line segment joining the points (3, −4), and (1, 2) is trisected at points P (a, −2) and Q $\left( \frac{5}{3}, b \right)$ , Then,

• $a = \frac{8}{3}, b = \frac{2}{3}$

• $a = \frac{7}{3}, b = 0$

• $a = \frac{1}{3}, b = 1$

• $a = \frac{2}{3}, b = \frac{1}{3}$

Q 41 | Page 66

f the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (−2, 5), then the coordinates of the other end of the diameter are

•  (−6, 7)

•  (6, −7)

•  (6, 7)

• (−6,−7)

Q 42 | Page 66

The coordinates of the point P dividing the line segment joining the points A (1, 3) and B(4, 6) in the ratio 2 : 1 are

• (2, 4)

• (3, 5)

•  (4, 2)

•  (5, 3)

Q 43 | Page 66

In Fig. 14.46, the area of ΔABC (in square units) is

• 15

• 10

• 7.5

• 2.5

Q 44 | Page 66

The point on the x-axis which is equidistant from points (−1, 0) and (5, 0) is

•  (0, 2)

•  (2, 0)

• (3, 0)

• (0, 3)

Q 45 | Page 66

If A(4, 9), B(2, 3) and C(6, 5) are the vertices of ∆ABC, then the length of median through C is

• 5 units

• $\sqrt{10}$ units

• 25 units

•  10 units

Q 46 | Page 66

If P(2, 4), Q(0, 3), R(3, 6) and S(5, y) are the vertices of a parallelogram PQRS, then the value of y is

• 7

• 5

• -7

• -8

Q 47 | Page 66

If A(x, 2), B(−3, −4) and C(7, −5) are collinear, then the value of x is

•  −63

• 63

• 60

•  −60

Q 48 | Page 66

The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is

• 7 + $\sqrt{5}$

• 5

• 10

• 12

Q 49 | Page 67

If the point  (2, 1 ) lies on the line segment joining points (4,20 and (8, 4) , then

• $AP = \frac{1}{3}AB$

• AP = BP

• PB =  $\frac{1}{3}AB$

• $AP = \frac{1}{2}AB$

Q 50 | Page 67

A line intersects the y-axis and x-axis at P and Q , respectively. If (2,-5) is the mid-point of PQ, then the coordinates of P and Q are, respectively

•  (0, -5) and (2, 0)

•  (0, 10) and ( - 4, 0)

• (0, 4) and ( -10, 0 )

• (0, - 0) and (4 , 0)

## Chapter 6: Co-Ordinate Geometry

Ex. 6.10Ex. 6.20Ex. 6.30Ex. 8.00Ex. 6.40Ex. 6.50Others

## RD Sharma solutions for Class 10 Mathematics chapter 6 - Co-Ordinate Geometry

RD Sharma solutions for Class 10 Maths chapter 6 (Co-Ordinate Geometry) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10 Mathematics chapter 6 Co-Ordinate Geometry are Centroid Formula, Co-ordinates of the Midpoint of a Segment, Section Formula, Division of a Line Segment, Distance Formula, Concepts of Coordinate Geometry, General Equation of a Line, Standard Forms of Equation of a Line, Intercepts Made by a Line, Slope of a Line, Area of a Triangle, Section Formula, Distance Formula, Graphs of Linear Equations, Concepts of Coordinate Geometry, Coordinate Geometry Examples and Solutions.

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