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The locus of the centre of a circle which touches the circle $\left|z-z_{1}\right|=a$ and $\left|z-z_{2}\right|=b$ externally $\left(z, z_{1}\right.$ and $z_{2}$ are complex numbers) will be

1) an ellipse
2) a hyperbola
3) a circle
4) none of these
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(2)

Explanation

Let the circle be $\left|z-z_{0}\right|=r .$ Then according
to given conditions $\left|z_{0}-z_{1}\right|=r+a$ and $\left|z_{0}-z_{2}\right|=r+b$. Eliminating $r$, we get $\left|z_{0}-z_{1}\right|-\left|z_{0}-z_{2}\right|=a-b .$
$\therefore$ Locus of centre $z_{0}$ is $\left|z-z_{1}\right|-\left|z-z_{2}\right|=a-b$, which represents a hyperbola.

by Diamond (75,948 points)

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