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  4. The locus of the centre of the circle which touch the circle |z - z1| = a and |z - z2| = b externally, where z, z1, z2 are complex numbers will be

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Q. The locus of the centre of the circle which touch the circle $|z - z_1| = a$ and $|z - z_2| = b$ externally, where $z, z_1, z_2$ are complex numbers will be

Complex Numbers and Quadratic Equations

Solution:

image
$|AP| = a + r$
$|BP| = b + r$
Now, $|AP| - |BP| = a - b$
$||AP|-|BP || = |a - b|$
Which represents a hyperbola.