A circle S = 0 touches a circle $x^{2} + y^{2} - 4 x + 6 y - 23 = 0$ internally and the circle $x^{2} + y^{2} - 4 x + 8 y + 19 = 0$ externally. The locus of centre of the circle S = 0 is conic whose eccentricity is k then $[\frac{1}{k}]$ is where [.] denotes G.I.F

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Solution

The correct option is A 7
$c_{1} (2, - 3), r_{1} = 6 c_{2} (2, - 4), r_{2} = 1$
Let C is the center of S = 0
$∴ \begin{matrix} c c_{1} = & r_{1} - r c c_{2} = & r_{1} + r \end{matrix}} \Rightarrow c c_{1} + c c_{2} = r_{1} + r_{2}$
$∴$ Locus is an ellipse whose foci are (2, -3) & (2, -4)
$e = \frac{2 a e}{2 a} = \frac{c_{1} c_{2}}{r_{1} + r_{2}} = \frac{1}{7} \Rightarrow k = \frac{1}{7}$

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