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 $[\cdot ]$ denotes greatest integer function)

Answer 1

Answer: 0

$C_1\left(-5,12\right)r_1=16$
$C_2\left(5,12\right)r_2=4$
Let $P$ be the centre of the third circle and $r$ be its radius. Then
$P{C}_1=16-r,P{C}_2=4+r$ then
$P{C}_1+P{C}_2=16-r+4+r=20$
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
$\frac{x^2}{100}+\frac{{\left(ax-12\right)}^2}{15}=1$
$D\ge 0\Rightarrow a^2\ge \frac{69}{100}$
$\Rightarrow \left[m\right]=0$