Question

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

• A. hyperbola
• B. ellipse
• C. auxiliary circle
• D. director circle

Answer 1

Answer: (D)

The equation of a pair of tangents from $\left(\alpha ,\beta \right)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is
$\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}-1\right)\left(\frac{{\alpha }^2}{a^2}+\frac{{\beta }^2}{b^2}-1\right)={\left(\frac{x\alpha }{a^2}+\frac{y\beta }{b^2}-1\right)}^2$
$\left(S{S}_1=T^2\right)$
The tangents are perpendicular, if the coefficient of $x^2+$ the coefficient of $y^2=0.$
$\Rightarrow \frac{1}{a^2}\left(\frac{{\alpha }^2}{a^2}+\frac{{\beta }^2}{b^2}-1\right)-\frac{{\alpha }^2}{a^4}+\frac{1}{b^2}\left(\frac{{\alpha }^2}{a^2}+\frac{{\beta }^2}{b^2}-1\right)-\frac{{\beta }^2}{b^4}=0$
$\Rightarrow \frac{{\beta }^2}{a^2{b}^2}-\frac{1}{a^2}+\frac{{\alpha }^2}{a^2{b}^2}-\frac{1}{b^2}=0$
$\Rightarrow {\alpha }^2+{\beta }^2=a^2{b}^2\left(\frac{1}{a^2}+\frac{1}{b^2}\right)$
$=a^2{b}^2\left(\frac{a^2+b^2}{a^2{b}^2}\right)<br/><br/>=a^2+b^2$
Hence, the locus of $\left(\alpha ,\beta \right)$ is the circle
$x^2+y^2=a^2+b^2$
This circle is called the director circle.