Question

Let $F_1$, $F_2$ be two foci of the ellipse and $LT$ and $LN$ be the tangent and the normal respectively to the ellipse at point $L$ then

• A. $LN$ bisects $\angle F_{1}L{F}_2$
• B. $LT$ bisects $\angle F_{1}L{F}_2$
• C. $LT$ bisects angle $({180}^{\circ }-\angle F_{1}L{F}_2)$
• D. none of these

Answer 1

Answer: (A), (C)

$LN$ bisects $\angle F_{1}L{F}_2$
$LT$ bisects angle $({180}^{\circ }-\angle F_{1}L{F}_2)$

Reflective property of ellipse: We know that the ray passing from $F_1$ after reflecting at any point on ellipse passes through $F_2$ i.e. the normal is angular bisector of $\angle F_{1}L{F}_2$.

i.e. $LN$ bisects $\angle F_{1}L{F}_2$

So as the another angular bisector must be perpendicular to it, it is the tangent. So tangent bisects the external angle i.e. $180°-\angle F_{1}L{F}_2$.

So $LT$ bisects $180°-\angle F_{1}L{F}_2$.