RD SHARMA Solutions for Class 10 Maths Chapter 6 - Co-ordinate Geometry
Chapter 6 - Co-ordinate Geometry Exercise Ex. 6.1
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Chapter 6 - Co-ordinate Geometry Exercise Ex. 6.2
Question 1 (i)
Solution 1 (i)
Question 1 (ii)
Solution 1 (ii)
Question 1 (iii)
Solution 1 (iii)
Question 1 (iv)
Solution 1 (iv)
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Show that the points A(1,-2), B(3,6), C(5,10) and D(3,2) are the vertices of a parallelogram.
Solution 7
Question 8
Prove that the points (1, 7), (4, 2), (-1, -1) and (-4, 4) are the vertices of a square.
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
The points A(2, 9), B(a, 5) and C(5, 5) are the vertices of triangle ABC right angled at B. Find the values of a and hence the area of ⧍ABC.
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Find a point which is equidistant from the points
A (-5, 4) and B (-1, 6). How many such points are there?
Solution 21
Question 22
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 
units.
Solution 22
Question 23
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.
Solution 23
Question 24
Solution 24
Question 25
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 (i) interior (ii) exterior of the triangle.
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29(i)
Show that the points A(5,6), B(1,5), C(2,1) and D(6,2) are the vertices of a square.
Solution 29(i)
Question 29(ii)
Prove that the points A(2, 3), B(-2, 2), C(-1, -2), and D (3, -1) are the vertices of a square ABCD.
Solution 29(ii)
Question 29(iii)
Solution 29(iii)
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
If the point P(x,y) is equidistant from the points A(5,1) and B(1,5), prove that x = y.
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
If the point P(k-1, 2) is equidistant from the points A(3, k) and B(k, 5), find the values of k.
Solution 36
Question 37
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.
Solution 37
Question 38
Solution 38
Question 39
Find the equation of the perpendicular bisector of the line segment joining points (7,1) and (3,5).
Solution 39
Question 40
Prove that the points (3,0), (4,5), (-1,4) and (-2,-1), taken in order, form a rhombus. Also, find its area.
Solution 40
Question 41
Solution 41
Question 42
Solution 42
Question 43
Solution 43
Question 44
If a point A(0,2) is equidistant from the points B(3,p) and C(p,5), then find the value of p.
Solution 44
Question 45
Prove that the points (7, 10), (-2, 5) and (3, -4) are the vertices of an isosceles right triangle.
Solution 45
Question 46
If the point P(x, 3) is equidistant from the points A(7, -1) and B(6, 8), find the value of x and the find the distance AP.
Solution 46
Question 47
If A(3, y) is equidistant from the points P(8, - 3) and Q(7, 6), find the value of y and find the distance AQ.
Solution 47
Question 48
If (0, -3) and (0, 3) are the two vertices of an equilateral triangle, find the coordinates of its third vertex.
Solution 48
For an equilateral triangle, the perpendicular bisector of any side passes through the opposite vertex.
Both the points, (0, -3) and (0, 3), lie on the y-axis equidistant from the origin. Hence, the perpendicular bisector joining these two points is the x-axis.
Any point on the x-axis has the coordinates (a, 0).
The distance between (0, -3) and (0, 3) is 6.
Hence, the distance between (a, 0) and (0, 3) should also be 6.
62 = (a - 0)2 + (0 - 3)2
36 = a2 + 9
a2 = 27
Question 49
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.
Solution 49
Question 50
Show that ∆ABC, where A (-2, 0), B (2, 0) C (0, 2) and ∆PQR, where P (-4, 0), Q (4, 0), R (0, 4) are similar.
Solution 50
Using the distance formula,
AB=
AC=
BC=
PQ=
PR=
QR=
Now,
ΔABC ~ ΔPQR
by the SSS test.
Question 51
Solution 51
Question 52
Find the circumcentre of the triangle whose vertices are (-2,-3), (-1,0), (7,-6).
Solution 52
Question 53
Solution 53
Question 54
Find the centre of the circle passing through (5, -8), (2, -9) and (2, 1).
Solution 54
Question 55
Solution 55
Question 56
Solution 56
Question 57
Solution 57
Chapter 6 - Co-ordinate Geometry Exercise Ex. 6.3
Question 1
Solution 1
Question 2
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).
Solution 2
(i) 
(ii) 
(iii) 
Question 3
Solution 3
Question 4
Solution 4
Question 5
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.
Solution 5
Question 6
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.
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
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.
Solution 9
Question 10
Solution 10
Question 11(i)
Solution 11(i)
Question 11(ii)
Solution 11(ii)
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
*Note: (i) Answer given in the book is incorrect.
Question 15
Solution 15
Question 16
Prove that (4,3), (6,4), (5,6) and (3,5) are the angular points of a square.
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
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.
Solution 19
Let the point on the x-axis be (a, 0).
Let this point divide the line segment AB in the ratio of r : 1.
Using the section formula for the y-coordinate, we get
Question 20
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.
Solution 20
Question 21
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.
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
In what ratio does the point (-4,6) divide the line segment joining the points A(-6,10) and B(3,-8)?
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Show that the points A(1,0) B(5,3), C(2,7) and D(-2,4) are the vertices of a parallelogram.
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37
Question 38
Points P, Q, R and S divide the line 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.
Solution 38
The difference between the x-coordinates of A and B is 6 - 1 = 5
Similarly, the difference between the y-coordinates of A and B is 7 - 2 = 5
Hence, if the line segment joining A(1, 2) and B(6, 7) is divided into 5 equal parts by the points P, Q, R and S, then the coordinates of P, Q, R and S can be found out by increasing the x and the y coordinates of A by 1 successively.
Hence, the coordinates of P are (1 + 1, 2 + 1) = (2, 3)
The coordinates of Q are (2 + 1, 3 + 1) = (3, 4)
The coordinates of R are (3 + 1, 4 + 1) = (4, 5)
Question 39
Solution 39
Question 40
Solution 40
Question 41
Solution 41
Question 42
Solution 42
Question 43
Solution 43
Question 44
Solution 44
Question 45
Solution 45
Question 46
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.
Solution 46
Question 47
If the coordinates of the mid-points of the sides of a triangle are (3,4), (4,6), and (5,7), find its vertices.
Solution 47
Question 48
Solution 48
Question 49
Solution 49
Question 50
Solution 50
Question 51
Solution 51
Question 52
Solution 52
Question 53
Solution 53
Question 54
Solution 54
Question 55
Solution 55
Question 56
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.
Solution 56
Given points are A(3,-5) and B(-4,8).
P divides AB in the ratio k : 1.
Using the section formula, we have:
Coordinate of point P are {(-4k+3/k+1)(8k-5/k+1)}
Now it is given, that P lies on the line x+y = 0
Therefore,
-4k+3/k+1 + 8k-5/k+1 =0
=> -4k+3+8k-5 =0
=> 4k -2 =0
=> k=2/4
=> k=1/2
Thus, the value of k is 1/2.
Question 57
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.
Solution 57
Question 58
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.
Solution 58
Question 59
ABCD is a parallelogram with vertices A(x1, y1), B(x2, y2) and C (x3, y3). Find the coordinates of the fourth vertex D in terms of x1, x2, x3, y1, y2 and y3.
Solution 59
Question 60
The points A (x1, y1), B(x2, y2) and C (x3, y3) are the vertices of ⧍ABC.
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
Solution 60
Chapter 6 - Co-ordinate Geometry Exercise Ex. 6.4
Question 1(i)
Solution 1(i)
Question 1(ii)
Solution 1(ii)
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Chapter 6 - Co-ordinate Geometry Exercise Ex. 6.5
Question 1
Solution 1
Question 2(i)
Solution 2(i)
Question 2(ii)
Solution 2(ii)
Question 2(iii)
Find the area of the quadrilaterals, the coordinates of whose vertices are
(-4,-2), (-3,-5), (3,-2), (2,3)
Solution 2(iii)
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Find the area of a quadrilateral ABCD, the coordinates of whose vertices are A(-3, 2), B(5, 4), C(7, -6) and D(-5, -4).
Solution 6
Question 7
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.
Solution 7
Question 8
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).
Solution 8
Question 9
If P(-5, -3), Q(-4, -6), R(2, -3) and S(1, 2) are the vertices of a quadrilateral PQRS, find its area.
Solution 9
Question 10
If A (-3, 5), B(-2, -7), C(1, -8) and D(6, 3) are the vertices of a quadrilateral ABCD, find its area.
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
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.
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
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 sequare units.
Solution 20
Question 21
If a ≠ b ≠ 0, prove that the points (a, a2), (b, b2), (0, 0) are never collinear.
Solution 21
Let the points (a, a2), (b, b2), (0, 0) represent a triangle. If we can prove that the area of the triangle so formed is not equal to zero, then we can prove that the points (a, a2), (b, b2), (0, 0) are never collinear.
Area of a triangle is given by
Now b≠a≠0.
So,
b - a≠0,
Δ≠0
Thus, points (a, a2), (b, b2), (0, 0) are never collinear.
Question 22
The area of a triangle is 5 sq. units. Two of its vertices are at (2, 1) and (3, -2). If the third vertex is (7/2, y), find y.
Solution 22
Area
of a triangle is given by
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Find the value (s) of k for which the points (3k - 1, k - 2), (k, k - 7) and (k - 1, -k - 2) are collinear.
Solution 30
Let the points be A, B and C respectively.
If A, B and C are collinear, then the area of ∆ABC is zero.
Question 31
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.
Solution 31
Question 32
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.
Solution 32
Question 33
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.
Solution 33
Question 34
A(6, 1), B(8, 2) and C(9, 4) are three vertices of a parallelogram ABCD. If E is the mid-point of DC, find the area of ⧍ADE.
Solution 34
Question 35
If D(-1/5, 5/2), E(7, 3) and F(7/2, 7/2) are the mid-points of sides of ⧍ABC, find the area of ⧍ABC
Solution 35
Chapter 6 - Co-ordinate Geometry Exercise 6.63
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
A line Segement 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
(a) 9,6
(b) 3,-9
(c) -3,9
(d) 9,-6
Solution 6
Question 7
Solution 7
So, the correct option is (d).
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Chapter 6 - Co-ordinate Geometry Exercise 6.64
Question 11
Solution 11
Question 12
If (-1,2),(2,-1) and (3,1) are any three vertices of a parallelogram, then
(a) a = 2, b = 0
(b) a = -2, b = 0
(c) a = -2, b = 6
(d) a = 6, b = 2
Solution 12
Note: The answer does not match the options.
Question 13
If A (5,3), B(11,-5) and P(12,y) are the vertices of a right triangle right angled at P, then y =
(a) - 2,4
(b) -2 ,4
(c) 2, -4
(d) 2,4
Solution 13
Question 14
The area of the triangle formed by (a,b+c), (b,c+a) and (c,a+b) is
(a) a+b+c
(b) abc
(c) (a+b+c)2
(d) 0
Solution 14
Question 15
If (x,2),(-3,-4) and (7,-5) are collinear, then x =
(a) 60
(b) 63
(c) -63
(d) -60
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
The line segment joining points (-3,4), and (1,-2) is divided by y - axis in the ratio
(a) 1 : 3
(b) 2 : 3
(c) 3 : 1
(d) 2: 3
Solution 20
Question 21
The ratio in which (4,5) divides the join of (2,3) and (7,8) is
(a) -2 : 3
(b) -3 : 2
(c) 3 : 2
(d) 2 : 3
Solution 21
Question 22
The ratio in which the x-axis divides the segment joining (3,6) and (12,-3) is
(a) 2:1
(b) 1 :2
(c) -2 : 1
(d) 1 : -2
Solution 22
Question 23
If the centrroid of the triangle formed by the points (a,b),(b,c) and (c,a) is at the origin, then a3 + b3 + c3 =
(a) abc
(b) 0
(c) a+b+c
(d) 3abc
Solution 23
Question 24
If points (1, 2), (-5, 6) and (a, -2) are collinear, then a =
(a) -3
(b) 7
(c) 2
(d) -2
Solution 24
Question 25
If the centroid of the triangle formed by (7, x), (y, -6) and (9, 10) is at (6, 3), then (x, y) =
(a) (4, 5)
(b) (5, 4)
(c) (-5, -2)
(d) (5, 2)
Solution 25
Chapter 6 - Co-ordinate Geometry Exercise 6.65
Question 26
The distance of the point (4, 7) from the x-axis is
(a) 4
(b) 7
(c) 11
(d) 
Solution 26
We know, distance of point (x, y) from x-axis is y.
Hence, distance of point (4, 7) from x-axis is 7.
Hence, correct option is (b).
Question 27
The distance of the point (4, 7) from the y-axis is
(a) 4
(b) 7
(c) 11
(d)
Solution 27
We know distance of point (x, y) from y-axis is x.
Hence distance of point (4, 7) from y-axis is 4.
Hence, correct option is (a).
Question 28
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
(a) (0, 3)
(b) (3, 0)
(c) (0, 0)
(d) (0, -3)
Solution 28
Given P is a point on x-axis
Hence P = (x, 0)
Distance from the origin is 3
Hence P = (3, 0)
Given Q is a point on y-axis
So Q is (0, y)
Given that OP = OQ
implies OQ = 3
Distance of Q from the origin is 3
Hence y = 3
implies Q = (0, 3)
Hence, correct option is (a).
Question 29
If the point (x, 4) lies on a circle whose centre is at the origin and radius is 5, then x =
(a) ±5
(b) ±3
(c) 0
(d) ±4
Solution 29
Question 30
If the point P (x, y) is equidistant from A(5, 1) and B (-1, 5), then
(a) 5x = y
(b) x = 5y
(c) 3x = 2y
(d) 2x = 3y
Solution 30
Question 31
If points A (5, p), B (1, 5), C (2, 1) and D (6, 2) form a square ABCD, then p =
(a) 7
(b) 3
(c) 6
(d) 8
Solution 31
Question 32
Solution 32
Question 33
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
(a) (0, 2)
(b) (3, 0)
(c) (0, 3)
(d) (2, 0)
Solution 33
Question 34
If the centroid of the triangle formed by the points (3, -5), (-7, 4), (10, -k) is at the point (k, - 1), then k =
(a) 3
(b) 1
(c) 2
(d) 4
Solution 34
Question 35
If (-2, 1) is the Centroid of the triangle having its vertices at (x, 2), (10, -2), (-8, y), then x, y satisfy the relation
a. 3x + 8y = 0
b. 3x - 8y = 0
c. 8x + 3y = 0
d. 8x = 3y
Solution 35
The Centroid of the triangle is given by
x=-8 and y=3 satisfy(a) 3x + 8y = 0.
Question 36
The coordinates of the fourth vertex of the rectangle formed by the points (0, 0), (2, 0), (0, 3) are
(a) (3, 0)
(b) (0, 2)
(c) (2, 3)
(d) (3, 2)
Solution 36
Question 37
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
(a) 3 or -9
(b) -3 or 9
(c) 6 or 27
(d) -6 or -27
Solution 37
Question 38
The ratio in which the line segment joining P(x1, y1) and Q(x2, y2) is divided by x-axis is
(a) y1 : y2
(b) -y1 : y2
(c) x1 : x2
(d) -x1 : x2
Solution 38
Question 39
The ratio in which the line segment joining points A(a1, b1) and B(a2, b2) is divided by y-axis is
(a) -a1 : a2
(b) a1 : a2
(c) b1 : b2
(d) -b1 : b2
Solution 39
Chapter 6 - Co-ordinate Geometry Exercise 6.66
Question 40
Solution 40
Question 41
If 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
(a) (-6, 7)
(b) (6, -7)
(c) (6, 7)
(d) -6, -7)
Solution 41
Question 42
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
(a) (2, 4)
(b) (3, 5)
(c) (4, 2)
(d) (5, 3)
Solution 42
Question 43
In the figure, the area of ΔABC (in square units) is
(a) 15
(b) 10
(c) 7.5
(d) 2.5
Solution 43
Question 44
The point on the x-axis which is equidistant from points (-1, 0) and (5, 0) is
(a) (0, 2)
(b) (2, 0)
(c) (3, 0)
(d) (0, 3)
Solution 44
Question 45
Solution 45
Question 46
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
(a) 7
(b) 5
(c) -7
(d) -8
Solution 46
Question 47
If A(x, 2), B (-3, -4) and C (7, -5) are collinear, then the value of x is
(a) -63
(b) 63
(c) 60
(d) -60
Solution 47
Question 48
Solution 48
Chapter 6 - Co-ordinate Geometry Exercise 6.67
Question 49
If the point P(2, 1) lies on the line segment joining points A (4, 2) and B (8, 4), then
Solution 49
Question 50
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
a. (0, -5) and (2, 0)
b. (0, 10) and (-4, 0)
c. (0, 4) and (-10, 0)
d. (0, -10) and (4, 0)
Solution 50
Other Chapters for CBSE Class 10 Mathematics
Chapter 1- Real Numbers Chapter 2- Polynomials Chapter 3- Pairs of Linear Equations in Two Variables Chapter 4- Quadratic Equations Chapter 5- Arithmetic Progressions Chapter 7- Triangles Chapter 8- Circles Chapter 9- Constructions Chapter 10- Trigonometric Ratios Chapter 11- Trigonometric Identities Chapter 12- Heights and Distances Chapter 13- Areas Related to Circles Chapter 14- Surface Areas and Volumes Chapter 15- Statistics Chapter 16- ProbabilityRD SHARMA Solutions for CBSE Class 10 Subjects
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