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(D) The equation of a pair of tangents from $(\alpha, \beta)$ to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is given by the condition $SS

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director circle

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(D) The equation of a pair of tangents from $(\alpha, \beta)$ to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is given by the condition $SS_{1}=T^{2}$.The tangents are perpendicular if the sum of the coefficients of $x^{2}$ and $y^{2}$ in the combined equation is zero.Expanding the condition,we get:$\left(\frac{1}{a^{2}}\right)\left(\frac{\alpha^{2}}{a^{2}}+\frac{\beta^{2}}{b^{2}}-1\right) - \frac{\alpha^{2}}{a^{4}} + \left(\frac{1}{b^{2}}\right)\left(\frac{\alpha^{2}}{a^{2}}+\frac{\beta^{2}}{b^{2}}-1\right) - \frac{\beta^{2}}{b^{4}} = 0$Simplifying this expression:$\frac{\beta^{2}}{a^{2}b^{2}} - \frac{1}{a^{2}} + \frac{\alpha^{2}}{a^{2}b^{2}} - \frac{1}{b^{2}} = 0$$\alpha^{2} + \beta^{2} = a^{2}b^{2}\left(\frac{1}{a^{2}} + \frac{1}{b^{2}}\right) = a^{2} + b^{2}$Thus,the locus of the point $(\alpha, \beta)$ is $x^{2} + y^{2} = a^{2} + b^{2}$,which is known as the director circle.

The locus of the point of intersection of perpendicular tangents to the ellipse is called

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