CBSE Class 10 Mathematics Coordinate Geometry Worksheet Set G

Question. The centre of a circle is C(2, k). If A(2, 1) and B(5, 2) are two points on its circumference, then the value of k is   
(A) 6
(B) 2
(C) –6
(D) –2

Answer: A

Question. The ratio in which the line joining (1, 3) an (2, 7) is divided by 3x + y = 9 is   
(A) 3 : 4
(B) 2 : 4
(C) 1 : 2
(D) 3 : 1

Answer: A

Question. The distance between the points (2k + 4, 5k) and (2k, –3 + 5k) in units is   
(A) 1
(B) 2
(C) 4
(D) 5

Answer: D

Question. The distance between the points (3k + 1, –3k) and (3k – 2, –4 –3k) (in units) is   
(A) 3k
(B) 5k
(C) 5
(D) 3

Answer: C

Question. The point which divides the line joining the points A(1, 2) and B(–1, 1) internally in the ratio 1 : 2 is _________.   
(A) (-1/5 ,5/3)
(B) (1/3 , 5/3)
(C) (–1, 5)
(D) (1, 5)

Answer: B

Question. The ratio in which the line joining (a + b, b + a) and (a – b, b – a) is divided by the point (a, b) is ___________.   
(A) b : a internally
(B) 1 : 1 internally
(C) a : b externally
(D) 2 : 1 externally

Answer: B

Question. If the line (3x -8y+5) +a(5x-3y +10) = 0 is parallel to X-axis, then a is   
(A) -(8/3)
(B) -(3/5)
(C) –2
(D) 1/2

Answer: B

Question. Find the area of a triangle formed by the lines 4x-y-8 = 0, 2x+y-10 = 0 and y = 0 (in sq units). 
(A) 5
(B) 6
(C) 4
(D) 

Answer: B

Question. Find the length of the longest side of the triangle formed by the line 3x + 4y =12with the coordinate axes.   
(A) 9
(B) 16
(C) 5
(D) 7

Answer: C

Question. Find the area of the triangle formed by the line 3x - 4y +12 = 0with the coordinate axes.   
(A) 6 units2
(B) 12 units2
(C) 1 units2
(D) 36 units2

Answer: A

Question. Find the equation of a line which divides the line segment joining the points (1, 1) and (2, 3) in the ratio 2 : 3 perpendicularly,   
(A) 5x - 5y + 2 = 0
(B) 5x + 5y + 2 = 0
(C) x + 2y - 5 = 0
(D) x + 2y + 7 = 0

Answer: C

Question. If A(-2,3) and B(2,3) are two vertices of ΔABCand G(0, 0) is its centroid, then the coordinates of C are 
(A) (0,-6)
(B) (-4,0)
(C) (4,0)
(D) (0,6)

Answer: A

Question. Let ΔABCbe a right angled triangle in which A(0, 2) and B(2, 0). Then the coordinates of C can be   
(A) (0, 0)
(B) (2, 2)
(C) either (A) or (B)
(D) none of these

Answer: C

Question. If ΔABCis a right angled triangle in which A(3, 0) and B(0, 5), then the coordinates of C can be   
(A) (5, 3)
(B) (3, 5)
(C) (0, 0)
(D) both (B) and (C)

Answer: D

Question. A triangle is formed by the lines x - y - 8,X-axis and Y-axis. Find its centroid.   
(A) (8/3 , 8/3 )
(B) (8, 8)
(C) (4, 4)
(D) (0, 0)

Answer: A

VERY SHORT ANSWER TYPE QUESTIONS

Question. If the distance between the points (x, 0) and (5, 8) is 10 units, find the value(s) of x.

Answer: –1 or 11

Question. What is the distance of the point A(3, –4) from y-axis?

Answer: 3 units

Question. In what ratio is the line joining the points P(7, 7) and Q(–4, 4) is divided by (0, –1)?

Answer: 4 : 7

Question. What are the coordinates of the centroid of triangle formed by points A(–2, 4), B(7, –3) and C(1, 5).

Answer: (2, 2)

Question. Find the coordinates of points which divides line joining (–4, 0) and (0, 6) in the ratio 1 : 3. 

Answer: (-3 , 3/2)

Question. Point A(3, –4) lies on circle of radius 5 cm with centre (0, 0). Write the coordinates of the other end of the diameter whose one end is A.

Answer: (–3, 4)

Question. Find the third vertex of a triangle if two of its vertices are (3, –6) and (–5, 2) and its centroid is at the point (2, 0).

Answer: (8, 4)

Question. What is the distance between the points (1, –2) and (–3, 2)?

Answer: 4√2 units

Question. What is the area of triangle formed by the points (–2, 0), (4, 0) and (2, 3).

Answer: 9 sq. units

Question. Find the mid-point of the line segment joining the points P(–3, 4) and Q (9, –6).

Answer: (3, –1)

Question. C is point on the perpendicular bisector of AB. What is the relation between A, B, C?

Answer: AC = BC

Question. What is the ordinate of any point on x-axis?

Answer: 0

Question. Find the coordinates of fourth vertex of the rectangle formed by the points (0, 0), (3, 0) and (0, 5).

Answer: (3, 5)

Question. What is the distance of the point (–6, 8) from origin?

Answer: 10 units

Question. What is the area of ΔABC, if points A, B and C are collinear?

Answer: zero

Question. In what ratio does the line segment joining the points (6, 4) and (1, –7) is divided internally by the axis of x?
Solution. 4 : 7

Question. Find the coordinates of the point of trisection of the line segment AB whose end points are A (2, 1) and B (5, –8).
Solution. (3, –2) and (4, –5)

Question. Find the length of median AD of a triangle whose vertices are A(–1, 3), B(1, –1) and C(5, 1).
Solution. 5 units

Question. Find the area of the quadrilateral, the coordinates of whose vertices are (1, 2), (6, 2), (5, 3) and (3, 4).
Solution. 11/2 sq.units

Question. What point on x-axis is equidistant from the points (–3, 4) and (7, 6)?
Solution. (3, 0)

Question. The coordinates of the centroid of a triangle are (1, 3) and the two vertices are (8, 5) and (–7, 6). Find the third vertex of the triangle.
Solution. (2, –2)

Question. If (3, 2), (4, 4) and (1, 3) are the mid-points of the sides of a triangle, find the coordinates of the vertices of the triangle.
Solution. (0, 1), (6, 3) and (2, 5)

Question. Three vertices of a parallelogram, taken in order are (3, 1), (2, 2) and (–2, 1) respectively. Find the coordinates of fourth vertex.
Solution. (–1, 0)

Question. Find the ratio in which the y-axis divides the segment joining (-3,6) and(12,-3).
Solution. 1/4

Question. Find the value of x for which the distance between the points P(4,-5) and Q ( 12 , x ) i s10 units.
Solution. 1, -11

Question. If the points A (4,3) and B(x,5) are on the circle with Centre O(2,3) then find the value of x.
Solution. 2

Question. What is the distance between the point A(c,0) and B(0,-c)?
Solution. √2 c

Question. For what value of p, are the points (-3,9),(2,p) and(4,-5) collinear?
Solution. -1

Question. Find the ratio in which the point P (x,2) divides the line-segments joining the points A(12,5)and B(4,- 3).Also ,find the value of x.
Solution. 3:5, x=9

Question. If the points A (-2, 1),B (a, b) and C(4,-1) are collinear and a-b=1.Find the value of a and b.
Solution. a=1, b=0

Question. In what ratio does the point (-4, 6) divides the line segment joining the points A (-6, 10)
Solution. 2/7

Question. Show that the points (3,2),(0,5),(-3,2) and (0,-1) are the vertices of a square.
Solution. Proof

Question. Point P divides the line segment joining the points A (2,1) and B(5,-8) such that AP:
AB=1:3 If P lies on the line2x-y+k=0, then find the value of k.
Solution. K=-8

Question. If the distance between the points (4,p) &(1,0) is 5, then find the value of p.
Solution. ±4

Question. If the point A(1,2), B(0,0) and C(a ,b) are collinear, then find the relation between a and b.
Solution. 2a=b

Question. 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.
Solution. 5:1, (0,-13/3)

Question. Find the distance between the points P (7,5) and Q(2,5).
Solution. 5

Question. If P (α/3 ,4) is the midpoint of the line segment joining the points Q(-6,5) and R( -2,3),then find the value of α.
Solution. -12

Question. By distance formula, show that the points (1,-1), (5,2) and (9,5) are collinear.
Solution. Proof

Question. Find the relation between x and y if the points(2,1),(x , y)and(7,5) are collinear
Solution. 4x - 5y - 3=0

Question. Find the ratio in which the line2x+3y=10 divides the line segment joining the points (1,2) and (2,3).
Solution. 2:3

Question. Find the coordinates of the point on y-axis which is nearest to the point (- 2, 5). 
Solution. The point on y-axis that is nearest to the point(-2,5) is (0,5).

Question. If 18, a ,b ,- 3 are in A.P., then find a + b. 
Solution. Since 18, a, b, and - 3 are in A.P., Then
a - 18 = - 3 - b
or, a + b = - 3 + 18
or, a + b = 15

PRACTICE EXERCISE

Question. A point P is at a distance of √13 units from the point (5, 4). Find the coordinates of P, if its ordinate is thrice of its abscissa.
Solution. (2,6) or (7/5 , 21/5)

Question. The area of a triangle is 5 square units. Two of its vertices are (2, 1) and (3, –2). The third vertex lies on y = x + 3. Find the third vertex.
Solution. (7/2 , 13/2) or ( -3/2 , 3/2)

Question. Show that the points A(2, –1), B(3, 4), C(–2, 3) and D(–3, –2) forms a rhombus but not a square. Find the area of the rhombus also.
Solution. 24 sq. units.

Question. Find the value of p if the distance between the points (3, p) and (4, 1) is 10 units.
Solution. p = 4 or –2

Question. Find the coordinates of the point which divides the line segment joining the points (4, –7) and (–5, 6) internally in the ratio 7 : 2.
Solution. (-3, -28/9)

Question. If the coordinates of two points A and B are (3, 4) and (5, –2) respectively. Find the coordinates of any point P, if PA = PB and area of ΔPAB = 10 square units.
Solution. (7, 2) or (1, 0)

Question. Find the coordinate of point P which divides the join of A(6, 5) and B(9, 2) in the ratio 1 : 2.
Solution. P(7, 4)

Question. If the coordinates of the mid-points of the sides of a triangle are (4, –3), (4, 5) and (–2, 3). Find the coordinates of its centroid.
Solution. (2 , 5/3)

Question. Find the area of the triangle whose vertices are :
(i) A (1, –1), B(–4, 6) and C(–3, –5) (ii) P (4, 2), Q(4, 5) and R (–2, 2)
(iii) A (1, 2), B(–2, 3) and C(–3, –4) (iv) P (5, 2), Q(4, 7) and R (7, –4)
Solution. (i) 24 sq. units (ii) 9 sq. units (iii) 11 sq. units. (iv) 2 sq. units

Question. A and B are the points (1, 2) and (2, 3). Find the coordinates of point C on the line segment AB such that 3AC = 4 BC.
Solution. C (11/7 ,18/7)

Question. Find the points of trisection of the line segment joining the points :
(i) (3, –2) and (–3, –4) (ii) (1, –2) and (–3, 4)
Solution. (i) (1 , -8/3 ) , (-1 , -10/3) (ii) (-1/3 , 0 ) , (-5/3 , 2)

Question. ABCD is a square with the opposite angular points A(3, 4) and C(1, –1). Find the coordinates of B and D.
Solution. (9/2 , 1/2) and (-1 /2 , 5/2)

Question. Find the coordinates of the circumcentre of a triangle whose vertices are A(5, 1), B(11, 1) and C(11, 9).
Solution. (8, 5)

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

Question. Find the value of y if the distance between the points (2, –3) and (10, y) be 10 units.
Solution. y = 3 or –9

Question. Find the ratio in which the point (11, 15) divides the line segment joining the points (15, 5) and (9, 20).
Solution. 2 : 1

Question. 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.
Solution. x = – 3 or 5

Question. Determine the ratio in which the line 3x + y – 9 = 0 divides the segment joining the points (1, 3) and (2, 7).
Solution. 3 : 4

Question. In what ratio the point (–3, k) divides the line segment joining the points (–5, –4) and (–2, 3). Hence, find the value of k.
Solution. 2 : 1; k = 2/3

Question. 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.
Solution. (i) 1 : 2, (1/3 , 0) (ii) 2 : 5; (0,-3/7)

Question. If the coordinates of the mid-points of the sides of a triangle are (1, 2), (0, –1) and (2, –1). Find the coordinates of its vertices.
Solution. (1, –4), (3, 2) and (–1, 2)

Question. Find the point on x-axis which is equidistant from the points (–4, 6) and (5, 9).
Solution.(3, 0)

Question. Find the point on the y-axis which is equidistant from the points (3, 2) and (–5, –2).
Solution. (0, –2)

Question. The coordinates of the middle points D, E, F of the sides BC, CA and AB respectively of a ΔABC are (–3, 2), (5, –7) and (11, 7) respectively, find the coordinates of the vertices A, B and C.
Solution. (19, –2), (3, 16), (–9, –12)

Question. The line segment joining the points (–6, 8) and (8, –6) is divided into four equal parts. Find the coordinates of the point of section.
Solution. (-5/2 , 9/2);(1, 1) ;(9/2 , -5/2)

Question. For what value of x, the distance between P(x, 7) and Q(–2, 3) is 4 5 units.
Solution. x = 6 or –10

Question. Find the ratio in which the line-segment joining the points (6, 4) and (1, –7) is divided internally by x-axis.
Solution. 4 : 7

Question. The line joining the points (2, 1) and (5, –8) is trisected at the points P and Q. If P lies on the line 2x – y + k = 0, find the value of k. 
Solution. k = – 8 or – 13

Question. The centre of a circle is (3k + 1, 2k – 1). If the circle passes through the point (–1, –3) and the length of its diameter be 20 units, find the value of k.
Solution. k = 2, - 46/13

Question. If the points (10, 5), (8, 4) and (6, 6) are the mid-points of the sides of a triangle, find its vertices.
Solution. (8, 7), (12, 3), (4, 5)

Question. Find the coordinates of the points on the x-axis which are at a distance of 5 units from the point (5, 4).
Solution. (8, 0) and (2, 0)

Question. Find the coordinates of the points on the y-axis which are at a distance of 13 units from the point (12, 9).
Solution. (0, 14) and (0, 4)

Question. Two vertices of a triangle are (1, 2) and (3, 5) and its centroid is at the origin. Find the coordinates of the third vertex.
Solution. (–4, –7)

Question. A (3, 2) and B(–2, 1) are two vertices of ΔABC whose centroid G has the coordinates (5/3 , -1/3) Find the coordinates of the third vertex C of the triangle.
Solution. (4, –4)

Question. If the coordinates of the mid-points of the sides of a triangle are (1, 1), (2, –3) and (3, 4). Find its centroid.
Solution. (2 , 2/3)

Question. Find the ratio in which the line segment joining the points (7, 3) and (–4, 5) is divided internally by y-axis.
Solution. 7 : 4

Question. Find the centre of a circle, the end points of whose one diameter are (–3, –1) and (5, 8).
Solution. (1 , 7/2)

Question. Find the lengths of medians of a ΔABC having the vertices A(5, 1), B(1, 5) and C(–3, –1).
Solution. AD = √37 , BE = 5, CF = 2√13

Question. The line segment joining the points (3, –4) and (1, 2) is trisected at the ponts 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.
Solution. p = 7/3 , q = 0

Question. (i) For what value of k, the points (k, –1), (5, 7) and (8, 11) are collinear?
(ii) For what value of k are the points (k, 2 – 2k), (–k + 1, 2k) and (–4 – k, 6 – 2k) lie on a straight line?
Solution. (i) k = – 1 (ii) k = - 1 or 1/2

Question. The vertices of some triangles are given below alongwith their areas. Find the value of a.
Vertices Area
(i) (2, 3), (6, –2), (–2, a) 6
(ii) (3, 8), (4, a), (5, –2) 8
Solution. (i) a = 5 or 11 (ii) a = – 5 or 11

Question. Find the distance between the points:
(i) A (–3, –2) and B (–6, –7)
(ii) P(a, 0) and Q(0, b)
(iii) A (–m, –n) and B (m, n)
(iv) R( 3 √1, 1) and S(0, 3)
(v) M(3√ 3, 3√ 3) and N(0, 0)
(vi) A (a sin α, – b cos α) and B (–a cos α, b sin α)
Solution.
(i) √34 units
(ii) √a2 + b2 units
(iii) √2 m2 + n2 units
(iv) 2√ 2 units
(v) 2 √6 units
(vi) √a2 + b2 (sin α + cosα)

Question. An equilateral triangle has two vertices at the points (0, 0) and (3, 3) . Find the coordinates of the third vertex.
Solution. (0, 2 √3) or (3, √ 3)

Question. Find the centroid of a triangle whose vertices are :
(i) (–2, 3), (2, –1), (4, 0) (ii) (4, –8), (–9, 7), (18, 13)
Solution. (i)(4/3 , 2/3) (ii) (13/3 , 4)

Question. Find the coordinates of the points equidistant from three given points A(5, 1), B(–3, –7) and C(7, –1).
Solution. (2, –4)

Question. Find the coordinates of a point whose distance from (3, 5) is 5 units and that from (0, 1) is 10 units.
Solution. (6, 9)

Question. A circle passes through the points A(3, 1), B(1, –3) and C(6, –8). Find the coordinates of the centre of the circle.
Solution. (6, –3)

Question. The coordinates of a vertex of a triangle are (2, 5) and the coordinates of the mid-points of the sides passing through this vertex are (8, 0) and (9, 3). Find the coordinates of the remaining vertices.
Solution. (14, –5) and (16, 1)

Question. Find the coordinates of the point of intersection of medians of ΔABC whose vertices are A(–7, 5), B(– 1, –3) and C(5, 7).
Solution. (–1, 3)

Question. Find the area of the quadrilateral, the coordinates of whose vertices are :
(i) (–3, 2), (5, 4), (7, –6) and (–5, –4) (ii) (–4, –2), (–3, –5), (3, –2) and (2, 3)
(iii) (–5, 7), (–4, –5), (–1, –6) and (4, 5) (iv) (6, 9), (7,4), (4, 2) and (3, 7)
Solution. (i) 80 sq. units (ii) 28 sq. units (iii) 72 sq. units (iv) 17 sq. units

Question. If the area of the quadrilateral whose angular points, taken in order, are (1, 2), (–5, 6), (7, –4), (p, –2) be zero, find the value of p.
Solution. p = 3

MCQ

1. Distance between the points (5, −3) and (8, 1) is

(a) 5 units (b) 6 units (c) 25 units (d) none of these

2. If the distance between (4, 0) and (0, x) is 5 units, then x is

(a) 2 (b) 3 (c) 4 (d) 5

3. Three points are collinear if they lie on a

(a) line (b) plane (c) both (a) & (b) (d) none of these

4. If the points (x, y), (2, 3) & (−3, 4) are collinear then

(a) x + y = 17 (b) x − y = 17 (c) x − 5y = 17 (d) x + 5y = 17

5. P is a point on x-axis at a distance 3 units from y-axis to its right the coordinates of P are

(a) (3, 0) (b) (0, 3) (c) (3, 3) (d) (−3, 3)

6. The distance of point A(4, −3) from the origin is

(a) 1 unit (b) 7 units (c) 5 units (d) 3 units

7. The co-ordinates of 2 points are (6, 0) & (0, 8). The co-ordinates of the midpoints are

(a) (3, 4) (b) (6, 8) (c) (0, 0) (d) (4, 3)

8. What point on the x-axis is equilistant from the point A(7, 6) & B(−3, 4) ?

(a) (0, 4) (b) (−4, 0) (c) (3, 0) (d) (0, 3)

9. X-axis divides the join of A(2, −3) & B(5, 6) in ratio

(a) 1 : 2 (b) 2 : 1 (c) 3 : 2 (d) 2 : 3

10. If the distance of P(x, y) from A(5, 1) and B(−1, 5) is same the which of the following is true

(a) 3x = 4y (b) x = 2y (c) 3x = 2y (d) x = 3y

11. If P(−1, 1) is the middle point of the line segment joining Q(−3, b) and R(1, b+4), then b is

(a) 1 (b) −1 (c) 2 (d) 0

12. The 3rd vertex of an equilateral triangle whose other & vertices are (1, 1) and (−1, −1) is

(a) ()3,3− (b) both a & b (c) ()3,3− (d) none of these

13. The centroid of the triangle having vertices (7, 5), (5, 7) and (−3, 3) is

(a) (3, −5) (b) (−3, 5) (c) (−3, −5) (d) (3, 5)

14. 2 vertices of ΔABC are A(−1, 4) & B(5, 2) and its centroid is G(0, −3). The coordinate of C are

(a) (4, 3) (b) (4, 15) (c) (−4, −15) (d) (−15, −4)

15. If the vertices of a triangle be (x1, y1), (x2, y2) and (x3, y3) then the coordinates of its centroid are

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