The centres of those circles which touch the circle x² + y² - 8x - 8y - 8 = 0 externally and also touch the x-axis, lie on:

Let C₁ → x² + y² - 8x - 8y - 8 = 0
Let a circle C₂ → (x - h)² + (y - k)² = r² touch the given circle
∴ r₁ = √g² + f² - c = √16 + 16 + 8 = √40 = 6
r₁ = 6
r₂ = r
(h - 4)² + (k - 4)² = (6 + r)²
C₂ touches x-axis, so k = r
(h - 4)² + (k - 4)² = (6 + k)²
h² - 8h + 16 + k² - 8k + 16 = 36 + k² + 12k
h² - 8h + 16 - 8k + 16 = 36 + 12k
h² - 8h - 20k - 4 = 0
(h - 4)² = 20k + 4
Replacing (h, k) by (x, y)
(x - 4)² = 4(5y + 1)
So, locus is a parabola.