Question
Question: The locus of the centre of a circle, which touches the circles \|z – z<sub>1</sub>\| = a and \|z – z...
The locus of the centre of a circle, which touches the circles |z – z1| = a and |z – z2| = b externally will be –
A
An ellipse
B
A hyperbola
C
A circle
D
None of these
Answer
A hyperbola
Explanation
Solution
Sol. Centre of the circle be z0 and radius r. Then its equation is |z – z0| = r circle (1) touches the circle |z – z1| = a externally.
Distance between centres = sum of radii
|z0 – z1| = a + r …(1)
|z0 – z2| = b + r …(2)
(1) – (2)
|z0 – z1| – |z0 – z2| = a – b
\ z0 lies on the curve |z – z1| – |z – z2| = a – b which is equation of a hyperbola.