Question

Circles $C_1$ and $C_2$, of radii $r$ and $R$ respectively, touch each other as shown in the figure. The line $l$ , which is parallel to the line joining the centres of $C_1$ and $C_2$, is tangent to $C_1$ at $P$ and intersects $C_2$ at $A,B$. If $R^2=2r^2$, then $\angle AOB$ equals

• A. $22\frac{1^{\circ }}{2}$
• B. $45^{\circ }$
• C. $60^{\circ }$
• D. $67\frac{1^{\circ }}{2}$

Answer 1

Answer: (B)

Given,

$O^{\prime }B=R$
$PM=BN=r$
Now, $R^2=2r^2$
$R=\sqrt{2}r$
$O^{\prime }B=\sqrt{2}r$
$BN=r$
$\therefore O^{\prime }N=r$
In $\Delta O^{\prime }BN$,
$O^{\prime }N=BN=r$
$\therefore \angle B{O}^{\prime }N=45^{\circ }$
Similarly, $\angle A{O}^{\prime }O=45^{\circ }$
$\therefore \angle A{O}^{\prime }B=90^{\circ }$
$\Rightarrow \angle AOB=\frac{1}{2}\angle A{O}^{\prime }E$
$\Rightarrow \angle AOB=45^{\circ }$