The locus of the centre of a circle which touches the circle $| z - z_{1} | = a$ and $| z - z_{2} | = b$ , externally (where $z_{1}, z_{2}, ϵ C)$ will be

an ellipse

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a hyperbola

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a circle

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None of these

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Solution

The correct option is C a hyperbola
$z_{1} z_{3} - z_{3} z_{2} = (a + r) - (b + r)$
$= a - b$
= a constant, which represent a hyperbola
Definition. A hyperbola is the locus of a point which moves in such a way that the difference of its distance from two fixed points (foci) is always constant.
Hence (B) is correct answer.

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The locus of the centre of a circle which touches the circles $| z - z_{1} | = a a n d | z - z_{2} | = b$ externally is

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