Question

If the curves, $x^2-6x+y^2+8=0$ and $x^2-8y+y^2+16-k=0, (k > 0)$ touch each other at a point, then the largest value of k is __________.
Given 859

Answer 1

Common tangent is $S_1 - S_2 = 0$
$\Rightarrow -6x + 8y - 8 + k = 0$
Use $p = r$ for $I^{st}$ circle
$\Rightarrow \frac{\left|-18-8+k\right|}{10} = 1$
$\Rightarrow k = 36$ or $16 \quad\Rightarrow k_{max} = 36$