CBSE Class 10 Maths HOTs Co-Ordinate Geometry

Question. If the point (x,y) is equidistant from the point(2,1) and(1,-2), then:
(a) x+3y = 0
(b)3x+y = 0
(c) x+2y = 0
(d) 3x+2y = 0
Answer : A

Question. The perimeter of the Triangle formed by the points (0,0), (2,0) and (0,2) is:
(a) 1- 2√2
(b) 2√2+1
(c ) 4+ √2
(d) 4 +2√2
Answer : D

Question. P (5,-3) and Q (3,y ) are the points of trisection of the line segment joining A (7,-2) and B(1,-5), then y equals:
(a) 2
(b) 4
(c) -4
(d) -5/2
Answer : C

Question. The x – Coordinate of a point P is twice its Y- Coordinate. If P is equidistant from Q (2,-5) and R(-3,6), Then the Coordinates of P are:
(a) (16,8)
(b) (14,7)
(c) (18,9)
(d) (10,5)
Answer : A

Question. Which of the points A(1,3), B(-3,2), C(3,4) and D(4,1) is nearest to the origin:
(a) A
(b) B
(c) C
(d) D
Answer : A

CASE STUDY - 1

CASE STUDY: CARTESIAN- PLANE

Using Cartesian Coordinates we mark a point on a graph by how far along and how far up it is.

The left-right (horizontal) direction is commonly called X-axis.

The up-down (vertical) direction is commonly called Y-axis.

In Green Park, New Delhi Ramesh is having a rectangular plot ABCD as shown in the following figure. Sapling of Gulmohar is planted on the boundary at a distance of 1m from each other. In the plot, Ramesh builds his house in the rectangular area PQRS. In the remaining part of plot, Ramesh wants to plant grass..

1. Find the coordinates of the midpoints of the diagonal QS
Answer : 6.5, 4)

2. Find the length and breadth of rectangle PQRS
Answer : 7cm & 4 cm

3. Find Area of rectangle PQRS.
Answer : 28 cm²

(Question of 1 Marks)

Question. Find the distance between the prints A(10 Cosθ ) and B(0,10 Sinθ)
Answer: 10

Question. Find the area of ΔABC where A (2,3), B (-2,1) and C (3,-2)
Answer: 11sq. units 

Question. Find the co-ordinates of the point which divides the line-segment joining the point (1,3) and (2,7) in the ratio 3:4
Answer: 10/7 , 33/7

Question. Find the area of the triangle formed by the points O (0,0), A(a,0) and B(0,h).
Answer: ab/2

Question. AB is the diameter of a circle whose centre is (2,-3). If the co-ordinates of A.B are (1,4), then find the co-ordinates of A.
Answer: (3,-10)

Question. Find the distance of the point (3,-4) from y-axis If A(3,2) , B(-2,1) are two vertices of ΔABC whose centroid G has the co-ordinates (5/3, 1/3). Find the co-ordinates of the third vertex C.
Answer: 3 units

Question. Find the co-ordinates of vertex B of the equilateral triangle OAB in the fig. 

Answer: (4,-2)

(Question of 2/3 Marks)

Question. Find the ratio in which the line-segment joining the points (-3,-4) and (1,-2) is divided by y-axis
Answer: 3:1

Question. Find the point of trisection of the line-segment joining the points (-3,4) and (1,-2).
Answer: (-5/3 , 2) , (-1/3 , 0)

Question. In the figure BOA is a right triangle and C is the midpoint of hypt AB.Show that it is equidistant from the vertices O,A and B. 

Answer: √a² + b² = AC = BC = OC

Question. If the area of a triangle formed by (x, 2x), (-2, 6) and (3,1) is 5 square units, then find the value of x.
Answer: x=2

Question. If the centroid of the triangle formed by the points (a,b), (b,c) and (c,a) at the origin, then find the value of a³+b³+c³
Answer: 3abc

Question. A point 𝒂 lies on the 𝑋 axis, What is the coordinate of this point ?
Answer: (𝑎,0)

Question. If the points A(x, 2), B(-3, 4) and C(7, -5) are collinear, then find the value of x
Answer : 𝑥= −63

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

Question. Find the point on the x-axis which is equidistant from points (−1,0) and (5,0)
Answer : (2, 0)

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

Question. What is the distance of the point P(2,3) from the X – axis?
Answer : 3 units

Question. In which quadrant the point P that divides the line segment joining the points A(2, -5) and B(5,2) in the ratio 2 : 3 lies?
Answer : 4th Quadrant

Question. If A and B are the points (−6,7) and (−1,−5)respectively, then find the distance 2AB.
Answer : 26

Question. If C (𝑎,3) is the mid-point of the line segment joining the points A(5,4) and B(−3,2),then find the value of 𝑎
Answer : 1

Question. Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (– 3, 4).
Answer : 3x + y – 5 = 0

Question. If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y
Answer : x = 6, y = 3

Question. Determine if the points (1, 5), (2, 3) and (-2, -11) are collinear.
Answer : Non- collinear

Question. If A (-2,1), B (a, 0), C (4, b) and D (1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides.
Answer : a = 1, b = 1
AB = CD = √10 units
BC = AD = √10 units

Question. Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.
Answer : (-1, 7/2) , (0, 5) , (1, 13/2)

Question. Determine the ratio in which the line 2x + y - 4 = 0 divides the line segments joining A (2, -2) and B (3,7). Q15. If Q (0, 1) is equidistant from P (5, – 3) and R (x, 6), find the values of x. Also find the distance QR and PR.
Answer : 2 : 9