The centres of those circles which touch the circle, x ^ 2 + y ^ 2...

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

has centre \left(4, 4\right)$and radius$6$.<br /><br />Let \left(h, k\right)$ be the centre of the circle which is touching the given circle externally



Then

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

$⇒h^{2} - 8h + 16 + k^{2} - 8k + 16 = 36 + 12k + k^{2}$

$⇒h^{2} - 8h - 20k - 4 = 0$ ,

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

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

Which is equation of parabola.