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प्रश्न
Find the coordinates of O, the centre passing through A( -2, -3), B(-1, 0) and C(7, 6). Also, find its radius.
उत्तर
let o (x, y) be the centre of the circle.
OA = OB (radii of square circle)
⇒ OA2 = OB2
(x + 2)2 + (y + 3)2 = (x + 1)2 + (y - 0)2
⇒ x2 + 4 + 4x + y2 + 9 + 6y = x2 + 1 + 2x + y2
⇒ 2x + 6y + 12 = 0
⇒ x + 3y + 6 = 0 .........(1)
Similarly, OB=OC
⇒ OB2 =OC2
(x + 1)2 + (y + 0)2 = (x + 7)2 + (y - 6)2
⇒ x2 + 1 + 2x + y2 = x2 + 49 - 14x + y2 + 36 +12y
⇒ 16x - 12y - 84 = 0
⇒ 4x - 3y - 21 = 0 .... (2)
⇒ 4x + 12y +24 = 0 .....(1)
on solving (1)& (2) we get,
- 15 y - 45 = 0
⇒ y = -3
from (1)
x - 9 + 6 = 0
⇒ x = 3
Thus coordinate of O are (3 , -3)
Radius =`sqrt ((3 + 2)^2 + (- 3 + 3)^2)`
`= sqrt (25 + 0)`
` = sqrt 25` units
= 5 units
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संबंधित प्रश्न
Prove that the points (0 , 0) , (3 , 2) , (7 , 7) and (4 , 5) are the vertices of a parallelogram.
Find the point on y-axis whose distances from the points A (6, 7) and B (4, -3) are in the ratio 1: 2.
The points A (3, 0), B (a, -2) and C (4, -1) are the vertices of triangle ABC right angled at vertex A. Find the value of a.
KM is a straight line of 13 units If K has the coordinate (2, 5) and M has the coordinates (x, – 7) find the possible value of x.
By using the distance formula prove that each of the following sets of points are the vertices of a right angled triangle.
(i) (6, 2), (3, -1) and (- 2, 4)
(ii) (-2, 2), (8, -2) and (-4, -3).
Show that the quadrilateral with vertices (3, 2), (0, 5), (- 3, 2) and (0, -1) is a square.
Find distance between point Q(3, – 7) and point R(3, 3)
Solution: Suppose Q(x1, y1) and point R(x2, y2)
x1 = 3, y1 = – 7 and x2 = 3, y2 = 3
Using distance formula,
d(Q, R) = `sqrt(square)`
∴ d(Q, R) = `sqrt(square - 100)`
∴ d(Q, R) = `sqrt(square)`
∴ d(Q, R) = `square`
The equation of the perpendicular bisector of line segment joining points A(4,5) and B(-2,3) is ______.
Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.
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Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane. |
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- Check if the Goal keeper G(–3, 5), Sweeper H(3, 1) and Wing-back K(0, 3) fall on a same straight line.
[or]
Check if the Full-back J(5, –3) and centre-back I(–4, 6) are equidistant from forward C(0, 1) and if C is the mid-point of IJ. - If Defensive midfielder A(1, 4), Attacking midfielder B(2, –3) and Striker E(a, b) lie on the same straight line and B is equidistant from A and E, find the position of E.
