Question

Circles $C_1$ and $C_2$ are externally tangent and they are both internally tangent to the circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10 , respectively and the center of the three circles are collinear. A chord of $C_3$ is also a common internal tangent of $C_1$ and $C_2$. Given that the length of the chord is $\frac{m\sqrt{n}}{p}$ where $m,n$ and $p$ are positive integers, $m$ and $p$ are relatively prime and $n$ is not divisible by the square of any prime, find the value of $(m+n+p-17)$.

Answer 1

Answer: 2

$\because \ell .\ell =8\times 20$
$\Rightarrow {\ell }^2=160$
$\Rightarrow \ell =4\sqrt{10}$
$\therefore 2\ell =8\sqrt{10}$
$\Rightarrow$ length of the chord $=\frac{8\sqrt{10}}{1}$
$=\frac{m\sqrt{n}}{p}$ (given)

where, $m,n,p\in N$
$\therefore m=8$
and $p=1$
Hence, $(m+n+p)=8+10+1=19$
$\therefore m+n+p-17=2$