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प्रश्न
Find the coordinate of O , the centre of a circle passing through A (8 , 12) , B (11 , 3), and C (0 , 14). Also , find its radius.
उत्तर
Let O (x , y) be the centre of the circle.
OA = OB (radii of the same circle)
⇒ OA2 = OB2
(x - 8)2 + (y - 12)2 = (x - 11)2 + (y -3)2
⇒ x2 + 64 - 16x + y2 + 144 - 24y = x2 + 121 - 22x + y2 + 9 - 6y
⇒ 6x - 18y + 78 = 0
⇒ x - 3y + 13 = 0
similarly , OB = OC
∴ OB2 = OC2
(x - 11)2 + (y - 3)2 = (x - 0)2 + (y - 14)2
⇒ x2 + 121 - 22x + y2 + 9 - 6y = x2 + y2 + 196 - 28y
⇒ - 22 x + 22 y - 66 = 0
⇒ - x + y - 3 =0 ..........(2)
x - 3y + 13 = 0 ..........(1)
solving (1) & (2) we get ,
- 2 y + 10 = 0
⇒ y = 5
from (1)
x - 15 + 13 = 0
⇒ x = 2
Thus , coordinates of O are (2,5)
Radius = `sqrt ((2 - 8)^2 + (5 - 12)^2) = sqrt (36 + 49) = sqrt 85` units
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संबंधित प्रश्न
The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q(2, –5) and R(–3, 6), find the coordinates of P.
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.
Find the distance of the following points from the origin:
(ii) B(-5,5)
Determine whether the points are collinear.
A(1, −3), B(2, −5), C(−4, 7)
Determine whether the point is collinear.
P(–2, 3), Q(1, 2), R(4, 1)
The length of line PQ is 10 units and the co-ordinates of P are (2, -3); calculate the co-ordinates of point Q, if its abscissa is 10.
Find the point on y-axis whose distances from the points A (6, 7) and B (4, -3) are in the ratio 1: 2.
Give the relation that must exist between x and y so that (x, y) is equidistant from (6, -1) and (2, 3).
Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
It is rectangular in shape - 100 yards by 60 yards. Goals consist of two upright posts placed equidistant from the centre of the backline, joined at the top by a horizontal crossbar. The inner edges of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the crossbar must be 2.14 metres (7 feet) above the ground.
Each team plays with 11 players on the field during the game including the goalie. Positions you might play include -
- Forward: As shown by players A, B, C and D.
- Midfielders: As shown by players E, F and G.
- Fullbacks: As shown by players H, I and J.
- Goalie: As shown by player K.
Using the picture of a hockey field below, answer the questions that follow:
The point on x axis equidistant from I and E is ______.
What is the distance of the point (– 5, 4) from the origin?
