Question 325.
If the centre of circle C2 of radius 10 lies on y = x and intersects a common chord of length 16 with the circle C1 having equation, x^2 + y^2 = 64, find the centre of the circle C2.
Answer 325.
Radius of the circle C1 : x^2 + y^2 = 64 is 8.
If the length of the common chord is 16, it is the diameter of the circle C1
=> it should pass through its center which is the origin.
If the center of the circle C2 lies on y = x, let it be (h, h)
=> Its equation is
(x - h)^2 + (y - h)^2 = (10)^2
=> x^2 + y^2 - 2hx - 2hy + 2h^2 - 100 = 0
=> Common chord of the circles C1 = 0 and C2 = 0 is
C1 - C2 = 0
=> (x^2 + y^2 - 64) - (x^2 + y^2 - 2hx - 2hy + 2h^2 - 100) = 0
=> 2h (x + y) - 2h^2 + 36 = 0
It passes through the origin,
=> - 2h^2 + 36 = 0
=> h^2 = 18
=> h = 3√2
=> center of C2 is (3√2, 3√2).
Link to YA!
If the centre of circle C2 of radius 10 lies on y = x and intersects a common chord of length 16 with the circle C1 having equation, x^2 + y^2 = 64, find the centre of the circle C2.
Answer 325.
Radius of the circle C1 : x^2 + y^2 = 64 is 8.
If the length of the common chord is 16, it is the diameter of the circle C1
=> it should pass through its center which is the origin.
If the center of the circle C2 lies on y = x, let it be (h, h)
=> Its equation is
(x - h)^2 + (y - h)^2 = (10)^2
=> x^2 + y^2 - 2hx - 2hy + 2h^2 - 100 = 0
=> Common chord of the circles C1 = 0 and C2 = 0 is
C1 - C2 = 0
=> (x^2 + y^2 - 64) - (x^2 + y^2 - 2hx - 2hy + 2h^2 - 100) = 0
=> 2h (x + y) - 2h^2 + 36 = 0
It passes through the origin,
=> - 2h^2 + 36 = 0
=> h^2 = 18
=> h = 3√2
=> center of C2 is (3√2, 3√2).
Link to YA!
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