Question

If two circles are given such that they neither intersect nor touches each other. Then the locus of centre of variable circle which touches both the circles externally is

• A. circle
• B. parabola
• C. ellipse
• D. hyperbola

Answer 1

The correct option is D hyperbola

Let the given two circles have centre $C_1$ and $C_2$ and radii as $r_1$ and $r_2$.
Let the circle which touches given two circles have centre at $C$ and radius $r$.

Now, $C{C}_2=r+r_2$ and $C{C}_1=r_1+r$
Hence, $C{C}_1-C{C}_2=r_1-r_2(=\text{constant})$
Hence locus of $C$ is hyperbola whose foci are at $C_1$ and $C_2$.