Question

Tangents are drawn from any point on the circle $x^2+y^2=R^2$ to the circle $x^2+y^2=r^2$. If the line joining the points of intersection of these
tangents with the first circle also touches the second circle, then $R=$

• A. $\frac{3r}{2}$
• B. $2r$
• C. $3r$
• D. $4r$

Answer 1

The correct option is B $2r$

From point 'P' on the circle
$x^2+y^2=R^2$, two tangents are drawn to the circle $x^2+y^2=r^2$
The tangents meet the first circle at points 'Q' and 'R'. Given that, QR is also a tangent to the circle $x^2+y^2=r^2$.
$PS=PT=\sqrt{R^2-r^2}$
Similarly, $SQ=QU=UR=TR=\sqrt{R^2-r^2}$
So, PQR is an equilateral triangle.
$2\theta ={60}^{\circ }$
$\theta ={30}^{\circ }$
In $△POS,\sin \theta =\frac{r}{R}$
$\sin {30}^{\circ }=\frac{r}{R}$
$\frac{1}{2}=\frac{r}{R}$
$R=2r$
'B' is the correct option.