Question

The locus of the centre of a circle which touches two given circles externally is:

• A. An ellipse
• B. A parabola
• C. A hyperbola
• D. None of these

Answer 1

Answer: (C)

A hyperbola

Let $S$ and $S^{\prime }$ be the centers and $a$ and $b$ be the radii of the given circles.

Let $P$ be the center and $r$ be the radius of the circle which touches the given circles externally.

Then $S^{\prime }P=r=a$ and $SP=r+b$

$\therefore S^{\prime }P-SP=(r+a)-(r+b)=a-b=$ constant

Hence the locus of $P$ is hyperbola whose foci are $S$ and $S^{\prime }$.