RD Sharma solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० chapter 6 - Co-Ordinate Geometry [Latest edition]

On which axis do the following points lie?

P(5, 0)

On which axis do the following points lie?

Q(0, -2)

On which axis do the following points lie?

R(−4,0)

On which axis do the following points lie?

S(0,5)

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.

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.

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.

Find the distance between the following pair of points:

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

Find the distance between the following pair of points:

 (a+b, b+c) and (a-b, c-b)

Find the distance between the following pair of points:

(asinα, −bcosα) and (−acos α, bsin α)

Find the distance between the following pair of points:

(a, 0) and (0, b)

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

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

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

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.

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

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

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

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

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.

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

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

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

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.

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

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

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.

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

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

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

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.

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

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

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

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

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

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.

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

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

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

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

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

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.

Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:

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

Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:

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

Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:

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

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

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

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?

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

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

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.

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

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.   

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. 

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

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.

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.

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

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

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

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

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

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

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

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

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

5, −6 and (−7, 5),

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

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

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

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

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

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

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 .

 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.

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.

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.

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 x, y and p. 

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.

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.

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

Find the ratio in which the point `P(3/4, 5/12)` divides the line segment joining the points `A(1/2, 3/2)` and B(2, –5). 

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.

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.

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

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

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

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

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.   

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.     

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.

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

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

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.

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

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.

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

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

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

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).

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

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

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

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.

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.

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.

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.   

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.

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

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

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.

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.

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

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

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.

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

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.

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

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.

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.

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.

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.

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(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

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.

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.

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.

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

 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  \[x_1 , x_2 , x_3 , y_1 , y_2 \text{ and }  y_3\]

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 A meets BC at D . Find the coordinates of the point  D.
(ii) Find the coordinates of the point P on AD such that AP : PD  = 2 : 1.
(iii) Find the points of coordinates Q and R 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 ? 

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

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

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

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.

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

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

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

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.

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

AB² + BC² + CA² = 3 (GA² + GB² + GC²)

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 (2a, 0) and C (0, 0).

Find the area of a triangle whose vertices are

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

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)`

Find the area of a triangle whose vertices are 

 (a, c + a), (a, c) and (−a, c − a)

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

 (−3, 2), (5, 4), (7, −6) and (−5, −4)

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

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

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

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.

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.

Show that the following sets of points are collinear. 

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

Show that the following sets of points are collinear. 

 (1, −1), (2, 1) and (4, 5)

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.

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).

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

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

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

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

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.     

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.

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

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.

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

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

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

If `a≠ b ≠ c`, prove that the points (a, a²), (b, b²), (c, c²) can never be collinear.

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

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).                 

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

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

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.      

If the points A(1, –2), B(2, 3) C(a, 2) and D(– 4, –3) form a parallelogram, find the value of a and height of the parallelogram taking AB as base.  

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

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  \[∆ ABC\] .

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

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.

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

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?

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

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

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

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?

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

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

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

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?

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.

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

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 .

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

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

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

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

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

If the distance between the points (4, p) and (1, 0) is 5, then p is equal to ______.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 =

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The ratio in which the line segment joining P (x₁, y₁) and Q (x₂, y₂) is divided by x-axis is

The ratio in which the line segment joining points A (a₁, b₁) and B (a₂, b₂) is divided by y-axis is

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,

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

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

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

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

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

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

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

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

If the point P(2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then ______.

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