Q: The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle x^{2} + y^{2} = 9 is

(A) (3/2 , 1/2)

(B) (1/2 , 3/2)

(C) (1/2 ,1/2)

(D) (1/2 , -√2)

Ans: (D)

Sol: Let the circle be given as x^{2} + y^{2} + 2gx + 2 fy + c = 0

This passes through (0,0) and (1,0). Therefore

c = 0 and 1 + 2g = 0

⇒ g = -1/2

It is given that the above circle touches x^{2 }+ y^{2} = 9. The centre of this circle, (0, 0) lies on the above given circle. From this it follows that the given circle touches internally the circle x^{2 }+ y^{2} = 9. Thus, the diameter of the required circle must be equal to the radius of the circle x^{2}+y^{2} = 9.

Hence, we have

$2 \sqrt{g^2 + f^2} = 3$

f = ±√2

Hence, centres of the required circle are (1/2 , ±√2)

Hence (D) is the correct answer.