Q. 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 $( m + n + p -17)$.
Conic Sections
Solution:
