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

Options

A. an ellipe which is not a circle
B. a hyperbola
C. a parabola
D. a circle

Answer

C

Explanation

Let C(h, k) be the centre of the circle touching x=axis and the circle `x^(2)+y^(2)-8x-8y-4=0` externally. The centre `C_(1)` of this circle has coordinates (4, 4) and its radius is 6.
The circles with centres at C and `C_(1)` touch each other externally.
`:." "C C_(1)=|k|+6`
`rArr" "sqrt((h-4)^(2)+(k-4)^(4))=|k|+6`
`rArr" "(h-k)^(2)+(k-4)^(2)=(|k|+6)^(2)`
`rArr" "(h-4)^(2)-8k=12|k|+20`
`rArr" "(h-k)^(2)=20k+20, if kgt0`
`"or, "(h-k)^(2)=-4k+20, ifklt0`
Hence, the locus of (h, k) is
`(x-4)^(2)=20(y+1)or,(x-4)^(2)=-4(y-5)`
which is a parabola.