Home
Class 12
MATHS
Two circles are given such that one is c...

Two circles are given such that one is completely lying inside the other without touching. Prove that the locus of the center of variable circle which touches the smaller circle from outside and the bigger circle from inside is an ellipse.

Text Solution

Verified by Experts

Let the circle `S_(2)` having centre at `C_(2)` and redius `r_(2)` lie completely insid the circle `S_(1)` having centre at `C_(1)` and radius `r_(1)`
Variable circle S touches `S_(1)` intrnally and `S_(2)` externally. Centre C of circle S is variable point and its radius 'r' is also variable. From the figure `C C_(2)=r+r_(2) and C C_(1) =r_(1)-r` So, `C C_(1)+ C C_(2)=r_(1)+r_(2)` (constant) Thus, locus of C is ellips whose foci are at `C_(1) and C_(2)`
Promotional Banner

Similar Questions

Explore conceptually related problems

The locus of the centre of the circles which touches both the axes is given by

Prove that the locus of the center of the circle which touches the given circle externally and the given line is a parabola.

The locus of the centre of a circle which touches two given circles externally is a

Two circles are given such that they neither intersect nor touch. Then identify the locus of the center of variable circle which touches both the circles externally.

The locus of the centre of a circle the touches the given circle externally is a _______

Prove that the locus of centre of the circle which toches two given disjoint circles externally is hyperbola.

The locus of the center of a circle which touches the circles |z-z_1|=a, |z-z_2=b| externally will be

Find the locus of centres of circles which touch two intersecting lines.

Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

The locus of centre of the circle touching x-axis nad the line y=x is