Question

From any point on the circle $x^2+y^2=a^2,$ two tangents are drawn to the circle $x^2+y^2=a^{2}si{n}^2\alpha ,$ then angle between them is

• A. $\frac{\alpha }{2}$
• B. $\alpha$
• C. $2\alpha$
• D. None of these

Answer 1

Answer: (C)

Let PQ and PR are two tangents and point P is
on the circle
$x^2+y^2=a^2$ with coordinates $(a\cos t,a\sin t)$ and $\angle OPQ=\theta$
Now, $PQ=$ length of tangent drawn from the point P to the circle $x^2+y^2=a^{2}si{n}^2\alpha$
$\therefore$ $PQ=\sqrt{a^2{\cos }^{2}t+a^2{\sin }^{2}t-a^2{\sin }^2\alpha }$
$=\sqrt{a^2-a^2{\sin }^2\alpha }$
$=a\cos \alpha$ and $OQ=$ radius of the circle $x^2+y^2=a^{2}si{n}^2\alpha$
$\Rightarrow$ $OQ=asin\alpha$
$\therefore$ $\tan \theta =\frac{OQ}{QP}=\frac{a\sin \alpha }{a\cos \alpha }=\tan \alpha$ $\theta =\alpha$
$\therefore$ Angle between both the tangents $=2\alpha$