If circles `x^(2)+y^(2)+2x+2y+c=0` and `x^(2)+y^(2)+2ax+2ay+c=0` where `c in R^(+), a != 1` are such that one circle lies inside the other, then

Options

A. `a in (0, sqrt((c)/(2)))-{1}`
B. `a in (- sqrt((c)/(2)),sqrt((c)/(2)))-{1}`
C. ` a in (-sqrt((c)/(2)),0)`
D. none of these

Answer

D

Explanation

If on circle lies inside the other circle, then
(i) both the circles will be lying on the same side of their radical axis and,
(ii) their radical axis will not intersect them.
The equation of the radical axis of two circles is
`(a-1)x+(a-1)y=0 rArr x+y=0 " " ...(i)`
Centre of the two circles are `C_(1)(-1, -1) and C_(2)(-a, -a)`. If points `C_(1) and C_(2)` lie on the same side of (i), then
`(-1-1) (-a-a) gt 0 rArr a gt 0`
The radical axis i.e. x+y=0 will not intersect the two circles, if the equation `2x^(2)+c=0` has imaginary roots. It is evidently true as ` c in R^(+)`.