Question

Let $P$ be any point on any directrix of an ellipse. Then chords of contact of point $P$ with respect to the ellipse and its auxiliary circle intersect at

• A. some point on the major axis depending upon the position of point $P$
• B. midpoint of the line segment joining the centre to the corresponding focus
• C. corresponding focus
• D. midpoint of the line segment joining the directrix to the corresponding focus

Answer 1

The correct option is C corresponding focus

Let the ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$
Any point on a directrix is $P(\frac{a}{e},k)$.
Chord of contact of $P$ with respect to the ellipse is
$\frac{a}{e}\cdot \frac{x}{a^2}+\frac{ky}{b^2}=1\dots \left(i\right)$
Auxiliary circle with respect to the ellipse is
$x^2+y^2=a^2$
Chord of contact of $P$ with respect to the auxiliary circle is
$\frac{a}{e}x+ky=a^2\dots \left(ii\right)$
Line $\left(i\right)$ and $\left(ii\right)$ intersect at $(ae,0),$ which is focus of the ellipse.