CBSE Class 12-science Answered

Dear Student,

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).

Regards Topperlearning.