Question

If two circles touch each other externally at the point $P$ and the straight line $l$ touches the two circles at the points $Q$ and $R$ respectively then $\angle QPR$ is equal to

• A. ${45}^{\circ }$
• B. ${55}^{\circ }$
• C. ${65}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is D ${90}^{\circ }$

Proof:
Let $\angle CAP=\alpha$ and $\angle CBA=\beta$
$CA=CP$ [length of the tangents from an eternal point C]
In a triangle $PAC,\angle CAP=\angle APC=\alpha$
Similarly $CB=CP$ and $\angle CPB=\angle PBC=\beta$
Now in the triangle $APB,$
$\angle PAB+\angle PBA+\angle APB=180$ [sum of interior angle in a triangle]
$\alpha +\beta +(\alpha +\beta )=180$
$2\alpha +2\beta =180\alpha +\beta =90$
$\therefore \angle APB=\alpha +\beta =90°$