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

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
\[
\left(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-1\right)\left(\frac{\alpha^{2}}{a^{2}}+\frac{\beta^{2}}{b^{2}}-1\right)=\left(\frac{x \alpha}{a^{2}}+\frac{y \beta}{b^{2}}-1\right)^{2}
\]
\(\left(\mathrm{SS}_{1}=\mathrm{T}^{2}\right)\)
The tangents are perpendicular, if the coefficient of \(x^{2}+\) the coefficient of \(y^{2}=0\).
\[
\begin{array}{r}
\Rightarrow \frac{1}{a^{2}}\left(\frac{\alpha^{2}}{a^{2}}+\frac{\beta^{2}}{b^{2}}-1\right)-\frac{\alpha^{2}}{a^{4}}+\frac{1}{b^{2}}\left(\frac{\alpha^{2}}{a^{2}}+\frac{\beta^{2}}{b^{2}}-1\right) \\
-\frac{\beta^{2}}{b^{4}}=0
\end{array}
\]
\[
\begin{gathered}
\Rightarrow \quad \frac{\beta^{2}}{a^{2} b^{2}}-\frac{1}{a^{2}}+\frac{\alpha^{2}}{a^{2} b^{2}}-\frac{1}{b^{2}}=0 \\
\Rightarrow \quad \alpha^{2}+\beta^{2}=a^{2} b^{2}\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}\right) \\
=a^{2} b^{2}\left(\frac{a^{2}+b^{2}}{a^{2} b^{2}}\right) \\
=a^{2}+b^{2}
\end{gathered}
\]
Hence, the locus of \((\alpha, \beta)\) is the circle
\[
x^{2}+y^{2}=a^{2}+b^{2}
\]
This circle is called the director circle.