Question

If r${}_1$ and r${}_2$ are the radii of smallest and largest circles which passes through $(5,6)$ and touches the circle $(x-2{)}^2+y^2=4$, then r${}_1$r${}_2$ is

• A. $\frac{4}{41}$
• B. $\frac{41}{4}$
• C. $\frac{5}{41}$
• D. $\frac{41}{6}$

Answer 1

The correct option is B $\frac{41}{4}$

Given circle is $(x-2{)}^2+y^2=4$
Centre is $(2,0)$ and radius $=2$
Therefore, distance between $(2,0)$ and $(5,6)$ is $\sqrt{9+36}=3\sqrt{5}$
$\Rightarrow r_1=\frac{3\sqrt{5}-2}{2}$
and $r_2=\frac{3\sqrt{5}+2}{2}$

Then, $r_1{r}_2=\frac{41}{4}$