First slide

Ellipse

Question

$Two circles are given such that one is completely lying inside the other without touching. Then the locus of$
$the centre of a variable circle which touches the smaller circle from outside and the bigger circle from inside is$

Moderate

A

An ellipse
B

A hyperbola
C

A parabola
D

A circle

Solution

$In the figure, circles with hard lines are the given$
$circles with centres C_{1} and C_{2} and radii r_{1} and r_{2} .$
$Let the circle with dotted line be the variable circle, which touches the given$
$two circles as erplained in the question, which has centre C and radius r .$
$\begin{array}{l} Now C C_{2} = r + r_{2} and C C_{1} = r_{1} - r \\ Hence, C C_{1} + C C_{2} = r_{1} + r_{2} (= constant) \end{array}$
$Then the locus of C is an ellipse whose foci are C_{1} and C_{2} .$

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