how to solve this question of parabola by only using euclidean geometry?

Solution 1:

This theorem is valid not only for a parabola but also for an ellipse or hyperbola. Let $F$ be a focus of a generic conic section, $A_1B_1$ and $A_2B_2$ two chords through $F$, $A_1'$ and $A_2'$ the projections of $A_1$ and $A_2$ on the directrix, $O$ the intersection between line $A_1A_2$ and the directrix.

From the similitude of triangles $OA_1A_1'$ and $OA_2A_2'$ one gets: $$ OA_1:OA_2=A_1A_1':A_2A_2'=FA_1:FA_2. $$ By the converse of exterior angle bisector theorem we then obtain that line $FO$ is the bisector of external angle $\angle A_1FB_2$ of triangle $FA_1A_2$.

The same reasoning can be repeated for triangle $FB_1B_2$: the line joining focus $F$ with the intersection of line $B_1B_2$ with the directrix is also the bisector of exterior angle $\angle B_2FA_1$. But those exterior angles coincide and have thus the same bisector: it follows that line $B_1B_2$ also intersects the directrix at point $O$. That completes the proof.

EDIT.

The argument given above can be expanded a bit, to prove some interesting properties of the tangents of a conic section.

If we let $A_2\to A_1$ in figure above, then lines $A_1A_2$ and $B_1B_2$ become the lines tangent to the conic at $A_1$ and $B_1$ respectively, endpoints of a focal chord (see figure below). The results proven above still hold, hence those tangents meet at a point $O$ on the directrix, and line $FO$ is the bisector of $\angle A_1FB_1$, i.e. $FO\perp A_1B_1$:

The line tangent to a conic section at a point $P$ meets the directrix relative to focus $F$ at a point $O$ such that $\angle PFO=90°$.

In the particular case of a parabola, it follows that triangles $A_1FO$ and $A_1A_1'O$ are congruent, implying that the line tangent at $A_1$ is the bisector of $\angle A_1'A_1F$, a first well known result:

The line tangent to a parabola at a point $P$ is the bisector of the angle formed by the line passing through $P$ and the focus, with the line through $P$ parallel to the axis.

Finally, from $\angle A_1'A_1F+\angle B_1'B_1F=180°$ one gets $\angle OA_1F+\angle OB_1F=90°$ and consequently $\angle A_1OB_1=90°$, which is another famous property:

The lines tangent to a parabola at the endpoints of a focal chord are perpendicular and meet on the directrix.