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

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, [Main 2016]
(a) a circle (b) an ellipse which is not a circle
(c) a hyperbola (d) a parabola
Ans: d
Sol.
The equation of circle is
x^2+y^2-8x-8y-4=0
⇒(x-4)^2+(y-4)^2=36=6^2
∴ Radius = 6.
Consider a line 6 units below the x-axis.

We have C^' C=C^' N=r+6
Thus, the locus of C^' is a parabola with C as focus and L as directrix.

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