Question

Let $C_1,C_2$ be two circles touching each other externally at the point A and let AB be the diameter of circle $C_1$. Draw a secant $B{A}_3$ to circle $C_2$, intersecting circle $C_1$ at a point $A_1(\ne A)$, and circle $C_2$ at points $A_2$ and $A_3$. If $B{A}_1=2$, $B{A}_2=3$ and $B{A}_3=4$, then the radii of circles $C_1$ and $C_2$ are respectively.

• A. $\frac{\sqrt{30}}{5},\frac{3\sqrt{30}}{10}$
• B. $\frac{\sqrt{5}}{2},\frac{7\sqrt{5}}{10}$
• C. $\frac{\sqrt{6}}{2},\frac{\sqrt{6}}{2}$
• D. $\frac{\sqrt{10}}{3},\frac{17\sqrt{10}}{30}$

Answer 1

The correct option is A $\frac{\sqrt{30}}{5},\frac{3\sqrt{30}}{10}$

Consider, $\Delta B{C}_{1}E$ and $\Delta B{C}_{2}F$

$\Rightarrow \angle E=\angle F={90}^{\circ }$

$\Rightarrow C_{1}E\parallel C_{2}F$

That means implies they are similar.

$\frac{B{C}_1}{B{C}_2}=\frac{BE}{BF}$

$\Rightarrow \frac{r_1}{r_1+r_1+r_2}=\frac{1}{3+\frac{1}{2}}$ .....$(\because \text{centre}{C}_1,C_2\text{bisect}B{A}_1,A_2{A}_3\text{with}\perp \text{at}E,F)$

$\Rightarrow 3r_1=2r_2$

$\Rightarrow r_2=\frac{3}{2}{r}_1$

Also if $P$ is an external point from which a secant cutting the circle at $A,B$, then $PA\times PB=\text{costant}$

Thus $B{C}_1\times B{C}_2=BE\times BF$

$\Rightarrow 4r_1(r_1+r_2)=4\times 3$

$\Rightarrow r_1^2+\frac{3}{2}{r}_1^2=4$

$\Rightarrow r_1^2=\frac{6}{5}$

$\Rightarrow r_1=\frac{\sqrt{30}}{5}$

And $r_2=\frac{3}{2}\frac{\sqrt{30}}{5}=\frac{3\sqrt{30}}{10}$