"Let S1 : x2 + y2 = 9 and S2 : (x $-$ 2)2 + y2 = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points :"

Question

Accepted Answer

"S₁ : x² + y² = 9 ; C₁ (0, 0), r₁ = 3

S₂ : (x $-$ 2)² + y² = 1 ; C₂ (2, 0), r₂ = 1

Image

Let the variable circle S and its radius is r units.

Here S and S₁ touches internally

$\therefore$ Distance between center,

S + S₁ = PC₁ = 3 $-$ r

Here S and S₂ touches externally

$\therefore$ Distance between center,

S + S₂ = PC₂ = 1 + r

$\therefore$ PC₁ + PC₂ = 4 > C₁ C₂

So locus is ellipse whose focii are C₁ & C₂ and major axis is 2a = 4 and 2ae = C₁C₂ = 2

$\Rightarrow$ $e = {1 \over 2}$

$\Rightarrow$ ${b^2} = 4\left( {1 - {1 \over 4}} \right) = 3$

Centre of ellipse is midpoint of C₁ & C₂ is (1, 0)

Equation of ellipse is $${{{{(x - 1)}^2}} \over {{2^2}}} + {{{y^2}} \over {{{\left( {\sqrt 3 } \right)}^2}}} = 1$$

Now by cross checking the option $\left( {2, \pm {3 \over 2}} \right)$ satisfied it."

Let S₁ : x² + y² = 9 and S₂ : (x $-$ 2)² + y² = 1. Then the locus of center of a variable circle S which touches S₁ internally and S₂ externally always passes through the points :

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Let S1 : x2 + y2 = 9 and S2 : (x $-$ 2)2 + y2 = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points :

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Let S₁ : x² + y² = 9 and S₂ : (x $-$ 2)² + y² = 1. Then the locus of center of a variable circle S which touches S₁ internally and S₂ externally always passes through the points :