Perpendicular tangents are drawn from the point P to an ellipse where axes are parallel to coordinate axes. The locus of point P is $x^{2} + y^{2} - 4 x - 6 y - 5 = 0$ and product of perpendicular from foci upon any one of these tangents is 6 then eccentricity is

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$\frac{1}{\sqrt{2}}$

$\frac{1}{\sqrt{3}}$

$\frac{1}{2}$

$\frac{1}{3}$

answer is D.

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Detailed Solution

locus of point of intersection of perpendicular tangents to an ellipse is director circle

$G i v e n d i r e c t o r c i r c l e i s {(x - 2)}^{2} + {(y - 3)}^{2} = 18$

$\Rightarrow a^{2} + b^{2} = 18$ and

product of perpendiculars from foci to any tangent is $b^{2}$

$b^{2} = 6 \Rightarrow a^{2} = 12$ then $e = \frac{1}{\sqrt{2}}$

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