Question

If from any point on the circle $x^2+y^2+2gx+2fy+c=0$ tangents are drawn to the circle $x^2+y^2+2gx+2fy+c{\sin }^2\alpha +(g^2+f^2){\cos }^2\alpha =0$, then the angle between the tangents is

• A. $2\alpha$
• B. $3\alpha$
• C. $\alpha$
• D. None of these

Answer 1

The correct option is A $2\alpha$

Clearly, the two circles are concentric.

For $C_1:$ center $=(-g,-f)$ and radius $=\sqrt{g^2+f^2-c}=r$

For $C_2:$ center $=(-g,-f)$ and

radius $=\sqrt{g^2+f^2-c{\sin }^2\alpha -g^2{\cos }^2\alpha -f^2{\cos }^2\alpha }=r\sin \alpha$

If $2\theta$ be the angle between the two tangents, then

$\sin \theta =\frac{r\sin \alpha }{r}=\sin \alpha \Rightarrow 2\theta =2\alpha$