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(C) The locus of the point of intersection of perpendicular tangents to an ellipse is known as the director circle.For the ellipse $\frac{x^2}{a^2} +

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$x^2 + y^2 = a^2 + b^2$

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(C) The locus of the point of intersection of perpendicular tangents to an ellipse is known as the director circle.For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,the equation of the director circle is given by $x^2 + y^2 = a^2 + b^2$.Alternatively,let the point of intersection be $(h, k)$. The pair of tangents from $(h, k)$ to the ellipse is given by $SS_1 = T^2$,where $S = \frac{x^2}{a^2} + \frac{y^2}{b^2} - 1$,$S_1 = \frac{h^2}{a^2} + \frac{k^2}{b^2} - 1$,and $T = \frac{hx}{a^2} + \frac{ky}{b^2} - 1$.Expanding this,the condition for perpendicular tangents is that the sum of the coefficients of $x^2$ and $y^2$ must be zero.This leads to $\frac{1}{a^2} (\frac{h^2}{a^2} + \frac{k^2}{b^2} - 1) - \frac{h^2}{a^4} + \frac{1}{b^2} (\frac{h^2}{a^2} + \frac{k^2}{b^2} - 1) - \frac{k^2}{b^4} = 0$,which simplifies to $h^2 + k^2 = a^2 + b^2$.Replacing $(h, k)$ with $(x, y)$,the locus is $x^2 + y^2 = a^2 + b^2$.

The locus of the point of intersection of perpendicular tangents to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is

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