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The line $y + 2 x + \sqrt{76}$ and 2y + x = 8 touch the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{12} = 1 .$ If the point of intersection of these two lines lies on a circle. Whose centre coincides with the centre of that ellipse, then the equation of that circle is

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$x^{2} + y^{2} = 16$

$x^{2} + y^{2} = 12$

$x^{2} + y^{2} = 28$

$x^{2} + y^{2} = {(y + \sqrt{8})}^{2}$

answer is C.

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Detailed Solution

Given two lines are perpendicular

point of inter section lies on director circle

$∴ eq u a t o n o f d i r e c t o r c i r c l e is x^{2} + y^{2} = 16 + 12 = 28$

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