Question

Find the equation of the tangents at the ends of the latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and also show that they pass through the points of intersection of the major axis and directions

Answer 1

$S:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Ends of latus rectum are $(ae,\frac{b^2}{a}),(ae,-\frac{b^2}{a})$

Equation of tangents will be $S_1=0$

$\frac{aex}{a^2}+\frac{b^{2}y}{a{b}^2}=1⟹ex+y=a$

$\frac{aex}{a^2}-\frac{b^{2}y}{a{b}^2}=1⟹ex-y=a$

The points of intersection will be $(\frac{a}{e},0)$

$⟹$ they pass through the points of intersection of the major axis and directrix.