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

Answer 1

Given circles are
$x^2-6x+y^2+8=0$ and $x^2-8y+y^2+16-k=0$
Centre and radius of the circle is
$C_1=(3,0),r_1=1C_2=(0,4),r_2=\sqrt{k}$
Two circles touch each other if
$C_1{C}_2=|r_1\pm r_2|$
$\Rightarrow \sqrt{k}+1=5$ or $|\sqrt{k}-1|=5$
$\Rightarrow$ $k=16$ or $36$

Hence, the largest value of $k$ is $36$.