Question

A circle passes through origin and has its centre on the line y = x. If the circle cuts the circle x² + y² - 4x - 6y +10 = 0 orthogonally, then its equation is

(A)  x² + y² + 2x + 2y = 0

(B)  x² + y² + 2x - 2y = 0

(C)  x² + y² - 2x - 2y = 0

(D)  x²  + y² - 2x - 2y = 0

Answer 1

Correct option  (D)  x²  + y² - 2x - 2y = 0

Explanation :

Let S≡ x² + y² + 2gx + 2fy + c = 0 be the required circle which passes through (0, 0). This implies that

c = 0  ....(1)

The circle has the centre on the line y = x which implies that

-g = -f     ....(2)

g = f   

The circle cuts the circle x² + y² - 4x - 6y + 10 = 0 orthogonally implies that

2(g)(-2) +2f(-3) = (-3) = c + 10

-4g -6f = c + 10  .....(3)

From Eqs. (1) – (3), we have g = f = -1 and c = 0. Therefore 

S ≡ x² + y² - 2x - 2y = 0