Let $\mathcal{E}$ be an ellipse, $\mathrm{F}$ be one of its foci and $l$ be the corresponding directrix. Let $\mathrm{AB}$ and $\mathrm{CD}$ be 2 focal chords through $\mathrm{F}$, such that $\mathrm{AD}$ and $\mathrm{BC}$ intersect at point $\mathrm{J}$. Then show that chord contact of $\mathrm{J}$ wrt $\mathcal{E}$ bisects the angle $\angle B F C$. As a corollary or otherwise, show that the perpendicular line from focus to any tangent, and the line joining the tangency point and the center of the ellipse intersect on the corresponding directrix:. (Geometrically)
i was able to show the first thing problems wanted : i used the fact that whenever pair of tangents are made at at the focal chords points on ellipse they intersect on directrix and the the line joining the point of intersection with the directrix with focus is the external angle bisector of a triangle.
From here i was able to show the below image and use the property that
- The line joining the intersection point of two tangent drawn to a focal chord is and the focus F is perpendicular to the focal chord :
But i was not able to get how by proving the first statement we can conclude anything about second statement which the problem asks ?
- Note : i used the converse that is if two focal chords AB,CD are given then the angle bisector line is the chord of contact of the point(J) which is the intersection of two chords AD,BC in the directrix. So i didn't rigorously prove the first statement I just tried proving the converse. .
Here's a possible approach.
If you let $A$ be the tangency point, let $C$ be the point on $\mathcal{E}$ such that $AC$ is a diameter of $\mathcal{E}$. Then the second statement should follow from the first.
A good reference for this kind of problem is Conic Sections Treated Geometrically, by W.H. Besant.
Almost straight out of the gate, Prop II on pg 6 speaks to the first statement.
Once you've drawn (as described above) the figure for the second statement, compare it to the figure for Prop XIX on pg 68.
I'm not sure how to get to the finish line, though. Maybe somebody can take it from here.