Question

The locus of the center of a circle, which touches externally the given two circle is

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

Answer 1

The correct option is C Hyperbola

Let $A$ and $B$ are the given two circles with radii $R_1$ and $R_2$ respectively and their centres are $F$ and $G$ respectively.

Let $C$ and $D$ are variable circles so that each circle meet the given circles $A$ and $B$ externally.

Let $r_1$ be the radius of circle $C$ and $P$ be its centre. Let $r_2$ be the radius of circle $D$ and $Q$ be its centre.

$PF=R_1+r_1$ , $PG=R_2+r_1$ , $PF-PG=R_1-R_2$

$QF=R_1+r_2$ , $QG=R_2+r_2$ , $QF-QG=R_1-R_2$

Hence locus of centres of touching circles is set of points so that the difference between the distances from two given fixed points ( here $F$ and $G$ ) to the point in locus is constant.

This locus is hyperbola and the fixed points $F$ and $G$ (Centers of given circles ) are foci of hyperbola