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
(a) An ellipse
(b) A hyperbola
(c) A parabola
(d) A circle

Answer 1

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Answer :-

⇝ The locus of variable circle's centre is ellipse.

Solution :-

Let's assume the Center and radius of Circles

⇝ Variable circle = centre (c ) , radius ( r)

⇝ Outer circle = center ( c1) , radius ( r1)

⇝ Inner circle = center ( c2) , radius ( r2)

⇝ Now find the distance between variable circle's centre and fixed circles centers .

⇝ Inner and outer circle are fixed . So there radius and centre are also fixed .

⇝ C C1 = Radius of (outer circle - variable circle)

⇝ C C1 = r1 - r .......... eq 1st

⇝ C C2 = Radius of ( Inner circle+ variable circle )

⇝ C C2 = r2 + r .......... eq 2nd

Adding both eq 1st and 2nd .

⇝ C C1 + C C2 = r1 - r + r2 + r

⇝ C C1 + C C2 = r1 + r2 = constant

( r1 and r2 are constant values )

I f if the sum of distance of a variable point from two fixed point is equal to constant then this will be an ellipse

Answer 2

Answer:

The locus of variable circle's centre is ellipse.

Solution :-

Let's assume the Center and radius of Circles

⇝ Variable circle = centre (c ) , radius ( r)

⇝ Outer circle = center ( c1) , radius ( r1)

⇝ Inner circle = center ( c2) , radius ( r2)

⇝ Now find the distance between variable circle's centre and fixed circles centers .

⇝ Inner and outer circle are fixed . So there radius and centre are also fixed .

⇝ C C1 = Radius of (outer circle - variable circle)

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⇝ C C1 = r1 - r .......... eq 1st

⇝ C C2 = Radius of ( Inner circle+ variable circle )

⇝ C C2 = r2 + r .......... eq 2nd

Adding both eq 1st and 2nd .

⇝ C C1 + C C2 = r1 - r + r2 + r

⇝ C C1 + C C2 = r1 + r2 = constant

( r1 and r2 are constant values )

I f if the sum of distance of a variable point from two fixed point is equal to constant then this will be an ellipse