The centres of those circles which touch the circle, $x^2+y^2-8x-8y-4=0,$ externally and also touch the $x$ - axis, lie on

Given circle $x^2+y^2-8x-8y-4=0$

has centre $\left(4, 4\right)$ and radius $6$.

Let $\left(h, k\right)$ be the centre of the circle which is touching the given circle externally



Then

$\sqrt{{\left(h-4\right)}^2+{\left(k-4\right)}^2}=6+k$

$\Rightarrow$$h^2-8h+16+k^2-8k+16=36+12k+k^2$

$\Rightarrow$$h^2-8h-20k-4=0$ ,

Replacing $h$ by $x$ and $k$ by $y$, locus is

$x^2-8x-20y-4=0$

Which is equation of parabola.