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Consider $S_{1} : x^{2} + y^{2} - 1 = 0$
$S_{2} : {(x - 8)}^{2} + {(y - 0)}^{2} = 25$
$S_{3} : {(x - 7)}^{2} + {(y - 1)}^{2} = 1$
Locus of centre of variable circle touching both $S_{1} & S_{2}$ externally is conic $C_{1}$
Locus of centre of variable circle touching both $S_{2} & S_{3}$ is conic $C_{2}$
Locus of centre of variable circle touching both $S_{1}$ & y-axis is conic $C_{3} (y > 0)$
Then choose correct option(s):-

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eccentricity of conic $C_{3}$ is 1

diameter of largest circle touches both $S_{2} & S_{3}$ is $6 + \sqrt{2}$

eccentricity of conic $C_{1}$ is 2

eccentricity of conic $C_{2}$ is $\frac{1}{3 \sqrt{2}}$

answer is A, B, C, D.

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Detailed Solution

$F o r C_{1} C C_{1} = r + 1, C C_{2} = r + 5 \Rightarrow C C_{2} - C C_{1} = 4 2 a = 4 & C_{1} C_{2} = 2 a e \Rightarrow e = 2$

$F o r C_{2} C C_{3} = r + 1, C C_{2} = 5 - r$

$C C_{2} + C C_{3} = 6$

$∴ 2 a = 6 & C_{2} C_{3} = 2 a e \Rightarrow a = 3 & e = \frac{1}{3 \sqrt{2}}$

For $C_{3}$ is parabola $∴ e = 1$
If S is largest circle as shown in the figure
its diameter $= 1 + \sqrt{2} + 5$ $= 6 + \sqrt{2}$

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