Question

Two circles are given such that one is completely lying inside the other without touching and their centers are not coincides. Then the locus of variable circle which touches the smaller circle externally and bigger circle internally, is

• A. Circle
• B. Parabola
• C. Ellipse
• D. Hyperbola

Answer 1

The correct option is C Ellipse


In the figure, we can see that the bigger circle has center $C_1$ and radius $r_1$ and the smaller circle has center $C_2$ and the radius $r_2$. Let the circle with the dotted line be a variable circle which touches the given two circles as explained in the question and has center C and radius r.
Now, ${\mathrm{CC}}_2=r_2+r\mathrm{and}{\mathrm{CC}}_1=r_1-r$
Hence, ${\mathrm{CC}}_1+{\mathrm{CC}}_2=r_1+r_2=\mathrm{constant}$
Then the locus of C is the ellipse whose foci are $C_1$ and $C_2$.