The locus of the centre of the circle which touches the two circles |z – z1| = a and |z – z2| = b is
(a) A circle
(b) A parabola
(c) A hyperbola
(d) An ellipse.
Let the center of the circle which touches both circles |z-z1|=a and |z-z2|=b be z0.Let the radius of this circle be r. So the equation of this circle becomes |z-z0|=r.
Now as this circle touches |z-z1|=a, therefore z1 is at a distance of (r+a) from z0.
=> |z0-z1|=a+r (1)
Similarly, as this circle also touches |z-z2|=b, therefore z2 is at a distance of (r+b) from z0.
=> |z0-z2|=b+r (2)
Subtracting (2) from (1), we get
|z0-z1| - |z0-z2| = a - b
Since z0 is a variable point, replace it by z and replace (a-b) by k.
=> |z-z1| - |z-z2| = k
This is a standard equation for a hyperbola. therefore, the locus of the centre of the circle which touches the given two circles is a hyperbola.
So, the correct answer is option (c).