R d sharma solutions mathematics class 10 Solutions for Chapter 13 co ordinate geometry

On which axis do the following points lie?

R (-4, 0)

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?

S (0, 5)

Find the distance between the following pair of points:

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

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

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

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 √10.

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

Prove that the points (2a, 4a), (2a, 6a) and (2a + √3a, 5a) are the vertices of an equilateral triangle.

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

Find the values of a and hence the area of triangle ABC.

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

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.

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

Find the fourth vertex.

Find a point which is equidistant from the points A (-5, 4) and B (-1, 6). How many such points are there?

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

Ayush starts walking from his house to office, Instead of going to the office directly, he goes to bank first, from there to his daughter’s school and then reaches the office. What is the extra distance travelled by Ayush in reaching the office? (Assume that all distance 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 kilometer.

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

If (-4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the interior.

If (-4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the exterior of the triangle.

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 circumcenter of the triangle whose vertices are (3, 0), (-1, -6) and (4, -1).

Also find the circumradius.

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

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 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 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 segment joining the points (-2, -3) and (3, 7) divides 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.

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

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

(2, – 2) and (-7, 4)

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

Find the ratio in which the line 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?

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.

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

Find the length 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 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 P(-1, y) lying on the line segment joining 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 center 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 the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also find the coordinates of the point of division.

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

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

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

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

Find the ratio in which the line 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 centroid of the triangle whose vertices are:

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