$\text{Two circles are given such that one is completely lying inside the other without touching. Then the locus of }$

$\mathrm{the} \mathrm{centre} \mathrm{of} a \mathrm{variable} \mathrm{circle} \mathrm{which} \mathrm{touches} \mathrm{the} \mathrm{smaller} \mathrm{circle} \mathrm{from} \mathrm{outside} \mathrm{and} \mathrm{the} \mathrm{bigger} \mathrm{circle} \mathrm{from} \mathrm{inside} \mathrm{is}$

$\text{In the figure, circles with hard lines are the given}$

$\text{ circles with centres }{C}_1\text{ and }{C}_2\text{ and radii }{r}_1\text{ and }{r}_2\text{ . }$

$\text{Let the circle with dotted line be the variable circle, which touches the given}$

$\mathrm{two} \mathrm{circles} \mathrm{as} \mathrm{erplained} \mathrm{in} \mathrm{the} \mathrm{question}, \mathrm{which} \mathrm{has} \mathrm{centre} C \mathrm{and} \mathrm{radius} r.$

$\begin{array}{l}\text{ Now }C{C}_2=r+r_2\text{ and }C{C}_1=r_1-r\\ \text{ Hence },C{C}_1+C{C}_2=r_1+r_2(=\text{ constant })\end{array}$

$\text{ Then the locus of }C\text{ is an ellipse whose foci are }{C}_1\text{ and }{C}_2\text{ . }$