Solution of Chapter 14. Co-ordinate Geometry (RD Sharma - Mathematics Book)

On which axis do the following points lie?

(i) P (5, 0) (ii) Q (0 -2)

(iii) R (- 4, 0) (iv) S (0, 5)

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when

(i) A coincides with the origin and AB and AD are along OX and OY respectively.

(ii) 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:

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

(ii) (a + b, b + c) and (a - b, c - b)

(iii) (a sin a, - b cos a) and (-a cos a, b sin a)

(iv) (a, 0) and (0, b)

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

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 A(- 4, -1), B(-2, - 4), C(4, 0) and D(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) and D (-4, 4) are the vertices of a square.

Prove that the points (3, 0), (6, 4) and (- 1, 3) are 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 a, 4 a), (2 a, 6 a) and are the vertices of an equilateral triangle.

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

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

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 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 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 circumcentre 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 (2, 1), (5, - 8) and (2, - 9).

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

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

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 that 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 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.

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.

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.

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

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

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

(iii) 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 points (3, 6) and (-3, 4).

If a point A (0, 2) is equidistant from the points B (3, p) and C (p, 5), then find the value of p.

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

If the point P (x, 3) is equidistant from the points A (7,-1) and B (6, 8), 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.

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

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

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:

(i) (5, -6) and (- 7, 5), (ii) (3, -2) and (-3, -4), (iii) (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.

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.

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 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.

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)?

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.

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

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

(i) x-axis

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 (-4, -1), (-2, -4), (4, 0) and (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).

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

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.

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.

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

Find the lengths of the medians of a ABC 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.

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.

If A and B are two points having coordinates (-2, -2) and (2, -4) respectively, find the coordinates of P such that AP = AB.

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

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

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

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

(iii) 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.

(iv) What do you observe?

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 mid-points of sides AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

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

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

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.

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

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).

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

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

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

The mid-point 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.

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

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

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.

Find the centroid of the triangle whose vertices are:

(i) (1, 4), (-1, -1), (3, -2)

(ii) (- 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.

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 + 3 GP².

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

AB² + BC² + CA² = 3 (GA² + GB² + GC²)

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

In Fig. 14.40, a right triangle BOA is given. C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices 0, A and B.

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.

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.

Find the area of a triangle whose vertices are

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

(ii)

(iii) (a, c + a), (a, c) and (-a, c - a)

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

(i) (-3, 2), (5, 4), (7, - 6) and (-5, - 4)

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

(iii) (-4, - 2), (-3, - 5), (3, - 2), (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 ∆ ABC 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.

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

(b) (1, -1), (2, 1) and (4, 5).

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

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 a ABC where B is (1, 5) and C (7, -2) is equal to 2 units.

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.

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

Four points A (6, 3), B (-3, 5), C (4, - 2) and D (x, 3x) are given in such a way that , find x.

For what value of a the 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 three points (x₁, y₁), (x₂, y₂), (x₃, y₃) lie on the same line, prove that

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

Find the value of k if points (k, 3), (6, - 2) and (-3, 4) are collinear.

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

If the point P (m, 3) lies on the line segment joining the points A( 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 k, if the points A (8, 1), B (3, - 4) and C (2, k) are collinear.

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 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 a parallelogram ABCD if three of its vertices are A(2, 4), B (2 + , 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) are collinear and a – b = 1, find the values of a and b.

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

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

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).