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

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 B a hyperbola
Let the circle |z-c|=d touch both the circles. We have to find the locus of the complex number c.
We have
$| c - z_{1} | = a + d a n d \Rightarrow | c - z_{2} | = b + d \Rightarrow | c - z_{1} | - | c - z_{2} | = a - b \Rightarrow | z - z_{1} | - | z - z_{2} | = a - b$
Thus difference between the distance of c from $z_{1} a n d z_{2}$ is constant.
Thus the locus of c is a hyperbola with foci $z_{1} a n d z_{2}$ .

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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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