Question

Let $C_{1}$ and $C_{2}$ are two circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$ respectively. Let $m$ be the smallest positive value of a for which the line $y=ax$ contains the centre of a circle that is internally tangent to $C_{1}$ and externally tangent to $C_{2}$ , then the value of $\left[m\right]$ is (where $\left[\right.\cdot \left]\right.$ denotes greatest integer function)

Answer 1

$C_{1}\left(- 5 ,12\right) \,\, r_{1}=16 $
$ C_{2}\left(5 ,12\right) \,\,\, r_{2}=4$
Let $P$ be the centre of the third circle and $r$ be its radius. Then
$PC_{1}=16-r,PC_{2}=4+r$ then
$PC_{1}+PC_{2}=16-r+4+r=20$
So, $P$ in on the ellipse centred and $\left(0 ,12\right)$ with foci $C_{1}$ and $C_{2}$ so major axis $=20$ and minor axis $=10\sqrt{3}$ . So equation of ellipse
$\frac{x^{2}}{100}+\frac{\left(y - 12\right)^{2}}{75}=1$
Now $y=ax$ satisfy it
$\frac{x^{2}}{100}+\frac{\left(a x - 12\right)^{2}}{15}=1$
$D\geq 0\Rightarrow a^{2}\geq \frac{69}{100}$
$\Rightarrow \left[m\right]=0$