Question

If from any point on the circle $x^2+y^2=a^2$ tangents are drawn to the circle $x^2+y^2=a^2\sin 2\alpha$ then the angle between the tangents, is

• A. $\alpha /2$
• B. $\alpha$
• C. $2\alpha$
• D. $4\alpha$

Answer 1

The correct option is C $2\alpha$

Let the point on circle $P(x_1,y_1)$

then ${x_1}^2+{y_1}^2={\theta }^2\left(1\right)$

We know, if $\theta$ is angle between tangent

then $\tan \left(\frac{\theta }{2}\right)=\frac{r}{\sqrt{S_{11}}}$

where $r=$ radius of circle on which tangents are drawn

$\sqrt{S_{11}}$ is the length of the tangent

then, $\tan \left(\frac{\theta }{2}\right)=\frac{\sqrt{a^2{\sin }^2\alpha }}{\sqrt{{x_1}^2+{y_1}^2-a^2{\sin }^2\alpha }}$

From $\left(1\right)$

$\tan \left(\frac{\theta }{2}\right)=\frac{a\sin \alpha }{\sqrt{a^2-a^2{\sin }^2\alpha }}=\frac{a\sin \alpha }{a\sqrt{1-{\sin }^2\alpha }}\tan \left(\frac{\theta }{2}\right)=\frac{\sin \alpha }{\cos \alpha }=\tan \alpha \frac{\theta }{2}=\alpha \theta =2\alpha$