Question

If the radical axis of the circles $x^2 + y^2 + 2gx + 2fy + c = 0$ and $2x^2 + 2y^2 + 3x + 8y + 2c = 0$ touches the circle $x^2 + y^2 + 2x + 2y + 1 = 0$ , then $(4g - 3)(f - 2) =$

• A. 0
• B. -1
• C. 1
• D. 2

Answer 1

Answer: (A)

The radical axis of the circles
$x^{2}+y^{2}+2 g x+2 f y+ c=0$
and $2 x^{2}+2 y^{2}+3 x+8 y+2 c=0$
is $(4 g-3) x+(4 f-8) y=0$ ...(i)
Since, the radical axis (i) touches the circle
$x^{2}+y^{2}+ 2 x+2 y+1=0$, so
$\frac{|-(4 g-3)-(4 f-8)|}{\sqrt{(4 g-3)^{2}+(4 f-8)^{2}}}=\sqrt{1+1-1}$
$\Rightarrow (4 g-3)^{2}+(4 f-8)^{2}+2(4 g-3)(4 f-8)$
$=(4 g-3)^{2}+(4 f-8)^{2}$
$\Rightarrow 8(4 g-3)(f-2)=0$
$\Rightarrow (4 g-3)(f-2)=0$