Question

The centre of a circle passing through the points (0, 0), (1, 0) touching the circle x² + y² = 9 is –

• A. (3/2, 1/2)
• B. (1/2, 3/2)
• C. (1/2, 1/2)
• D. (1/2, –2^1/2)

Answer 1

Answer: (D)

(1/2, –2^1/2)

Answer 2

C₁ = x² + y² – 9 = 0 ; C₂ = x² + y² + 2gx + 2fg + c = 0

passes (0,0) ̃ c = 0

̃ passes (1, 0) ̃ 1 + 2g = 0 ̃ g = –1/2

C₂(–g, –f) ; (r₂ =$\sqrt{g^{2} + f^{2} - c}$)

C₂(1/2, –f) ; r₂ =$\sqrt{1/4 + f^{2}}$

touches internally

C₁C₂ = |r₁ – r₂|

$\sqrt{1/4 + f^{2}}$ = 3 –$\sqrt{1/4 + f^{2}}$

2$\sqrt{1/4 + f^{2}}$= 3 ̃ 4(1/4 + f²) = 9 ̃ f = ±$\sqrt{2}$