Question

Locus of the point of intersection of the tangents at the end points of a focal chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

• A. auxiliary circle of the ellipse
• B. ellipse
• C. director dircle of ellipse
• D. directrices of ellipse

Answer 1

The correct option is D directrices of ellipse

Let point of intersection be $(h,k).$

Equation of chord of contact from $(h,k)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with $(a>b)$ is :
$\frac{hx}{a^2}+\frac{ky}{b^2}=1.$

Since, it is a focal chord, so it will pass through the focus $(\pm ae,0)$
$\Rightarrow \pm \frac{he}{a}+0=1\Rightarrow h=\pm \frac{a}{e}$

Hence, required locus is $x=\pm \frac{a}{e},$ which are pair of straight lines representing the directrices to ellipse.