R.D. Sharma solutions for मैथमैटिक्स [अंग्रेजी] कक्षा 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:

(a sin α, –b cos α) and (–a cos α, b sin 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`.

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 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.

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

Show that the points (–2, 3), (8, 3) and (6, 7) are the vertices of a right-angled triangle.

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 (2, 3), (-4, -6) and `(1, 3/2)` do not form a triangle.

The points A(2, 9), B(a, 5) and C(5, 5) are the vertices of a triangle ABC right angled at B. Find the values of a and hence the area of ∆ABC.

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.

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

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

If the point A(2, – 4) is equidistant from P(3, 8) and Q(–10, y), find the values of y. Also find distance PQ.

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

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

A (2, –2), В (7, 3), С (11, –1), D (6, –6)

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)

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

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

Name the type of triangle PQR formed by the points `P(sqrt(2), sqrt(2)), Q(-sqrt(2), -sqrt(2))` and `R(-sqrt(6), sqrt(6))`.

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.

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.

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

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.

Prove that the abscissa of a point P, which is equidistant from points with coordinates A(7, 1) and B(3, 5) is 2 more than its ordinate.

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

The centre of a circle is (2a, a – 7). Find the values of a if the circle passes through the point (11, –9) and has diameter `10sqrt(2)` units.

Ayush starts walking from his house to office. Instead of going to the office directly, he goes to a bank first, from there to his daughter’s school and then reaches the office. What is the extra distance travelled by Ayush in reaching his office? (Assume that all distances covered are in straight lines). If the house is situated at (2, 4), bank at (5, 8), school at (13, 14) and office at (13, 26) and coordinates are in km.

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

If (–5, 3) and (5, 3) are two vertices of an equilateral triangle, then find the coordinates of third vertex, given that origin lies inside the triangle. `("Take"  sqrt(3) = 1.7)`

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).

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 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.

Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b.

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.

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.

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

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.   

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.

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.

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.

ABCD is a rectangle formed by joining 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.

ABCD is a rectangle formed by joining the points A(–1, –1), В(–1, 4), С(5, 4) and D(5, –1). P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively. Show that the diagonals of the quadrilateral PQRS bisect each other.

Find the ratio in which the point P(–1, y) lying on the line segment joining points A(–3, 10) and B(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).

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 ratio in which the line segment joining the points A(6, 3) and B(–2, –5) is divided by x-axis.

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

P(–2, 5) and Q (3, 2) are two points. Find the coordinates of the point R on the line segment PQ such that PR = 2QR.

Find the coordinates of the point C, which lies on the line AB produced such that AC = 2BС, where the coordinates of points A and B are (–1, 7) and (4, –3), respectively.

Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, –3). Hence, find m.

Find the ratio in which the point `(8/5, y)` divides the line segment joining the points (1, 2) and (2, 3). Also, find the value of y.

Find the coordinates of the point R on the line segment joining the points P(–1, 3) and Q(2, 5) such that PR = `3/5` PQ.

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

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.

If A and B are two points having coordinates (–2, –2) and (2, –4) respectively, find the coordinates of P such that `AP = 3/7 AB`. Also, find the coordinates of P on AB such that `BP = 4/7 AB`.

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 A is nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.

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.

Find the ratio in which the line 2x + 3y – 5 = 0 divides the line segment joining the points (8, –9) and (2, 1). Also find the coordinates of the point of division.

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 the 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 (–1, 2) 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  \[x_1, x_2, x_3, y_1, y_2 \text{ and }  y_3\].  

If the points A(6, 1), B(p, 2), C(9, 4) and D(7, q) are the vertices of a parallelogram ABCD, then find the values of p and q. Hence, check whether ABCD is a rectangle or not.

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 that BQ : QE = 2 : 1 and CR : RF = 2 : 1.

(iv) What are the coordinates of the centroid of the triangle ABC? 

A (4, 2), В (6, 5) and C (1, 4) are the vertices of ΔABC.

1. The median from A meets BC in D. Find the coordinates of the point D. 
2. Find the coordinates of point P on AD such that AP : PD = 2 : 1. 
3. Find the coordinates of the points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1. 
4. What do you observe?

Find the length of the median through the vertex B of ΔABC with vertices A(9, −2), B(−3, 7) and C(−1, 10).

If the mid-point of the line segment joining the points A(3, 4) and B(k, 6) is P(x, y) and x + y – 10 = 0, find the value of k.

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

Find the centroid of the triangle whose 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.

Find the area of ΔABC whose vertices is A(–5, 7), B(–4, –5) and C(4, 5).

Find the area of a triangle whose vertices is (1, –1), (–4, 6) and (–3, –5).

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 quadrilateral whose vertices, taken in order, are (–4, –2), (–3, –5), (3, –2) and (2, 3).

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.

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.

Find the value of k so that the area of the triangle with vertices A(k + 1, 1), B(4, –3) and C(7, –k) is 6 square units. 

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 the point (a, 1), (1, –1) and (11, 4) are collinear?

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.  

Find the area of triangle whose vertices are `(at_1^2, 2at_1), (at_2^2, 2at_2)` and `(at_3^2, 2at_3)`.

Find the area of triangle whose vertices are (a, c + a), (a, c) and (–a, c – a).

In ∆ABC, the coordinates of vertex A are (0, –1) and D (1, 0) and E (0, 1)  respectively the mid-points of the sides AB and AC. If F is the mid-point of 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 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.

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 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 \[\frac{y_2 - y_3}{x_2 x_3} + \frac{y_3 - y_1}{x_3 x_1} + \frac{y_1 - y_2}{x_1 x_2} = 0\].

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 \[∆ ABC\].

Find the distance of a point P(x, y) from the origin.

Write the coordinates of a point P on x-axis which is equidistant from the point A(–2, 0) and B(6, 0).

Find the value(s) of x, if the distance between the points A(0, 0) and B(x, –4) is 5 units.

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

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

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.

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?

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

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

If the line joining the points A(4, –5) and B(4, 5) is divided by the point P such that `(AP)/(AB) = (2)/(5)`, find the coordinates of P.

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

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

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

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

Find the type of triangle ABC formed whose vertices are A(1, 0), B(−5, 0) and C(−2, 5).

In what ratio is the line segment joining the points (3, −5) and (−1, 6) divided by the line y = x? 

A(3, 0), B(6, 4) and C(−1, 3) are vertices of a triangle ABC. Find length of its median BE. 

Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y). Find the values of y. Hence find the radius of the circle.

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 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?

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

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

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?

In Q. No. 33, what is the value of \[\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}\]?

What are the coordinates of the point where the perpendicular bisector of the line segment joining the points A (1, 5), and B (4, 6) cuts the y-axis?

The distance of the point (2, 3) from x-axis is ______.

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

If the centroid of the triangle whose vertices are (2, 4), (3, a), (4, 2) is (a, 3), then a = ______.

The distance between the points (1, 0) and (2, cot θ) is ______.

The distance between the points (0, 5) and (5, 0) is ______.

If the distance between the points (2, –2) and (–1, x) is 5, then the values of x are ______.

The x-axis divides the line segment joining the points (–4, –6) and (–1, 7) in the ratio ______.

The quadrant in which the point dividing the line segment joining the points (7, –6) and (3, 4) internally in the ratio 1 : 2 is ______.

If the distance between the points (4, p) and (1, 0) is 5, then the values of p are ______.

If `P(a/3, 4)` is the mid-point of the line segment joining the points Q(–6, 5) and R(–2, 3), then the value of 'a' is ______.

The values of y for which the point (2, –4) is equidistant from the points (3, 8) and (–10, y), are ______.

The image of the point (3, –5) in the x-axis has the coordinates ______.

The coordinates of the image of the point (–4, 5) in y-axis are ______.

The x-coordinate of the point lying on the perpendicular bisector of the line segment joining the points A(–2, –5) and B(2, 5) is ______.

If the distance between the points A(–3, –14) and B(a, –5) is 9 units, then a = ______.

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) is ______.

The number of points on x-axis which are at a distance of `2sqrt(5)` from the point (7, 4), is ______.

It (2, –2) and (5, 2) are opposite vertices of a square, then the length of the side of the square is ______.

The coordinates of the point which is equidistant from the vertices of the triangle formed by the points O(0, 0), A(a, 0) and B(0, b), are ______.

If the distance of the point (4, a) from x-axis is half its distance from y-axis, then ______.

The coordinates of the point equidistant from the vertices O(0, 0), A(6, 0) and B(0, 8) of ΔOAB are ______.

If the centroid of the triangle formed by the points (a, b), (1, a) and (b, 1) is at the origin, then `(a^3 + b^3 + 1)/(ab)` = ______.