If parabola with focus (2/5,4/5)...

Question

so I was doing this question And decided to go about it as shown below. Everything was going smooth until I came to eq. no 2 . here even though when we built the equation using all the correct things the eq 2 gives two roots where as it should have been a whole square in order to give us a single x-coordinate for A . Secondly if you all think that there would be a better way to solve this question ,please share it . It will help me gain better and deeper knowledge.

Answer 1

The line tangent to a parabola at any point $P$ is the bisector of the angle formed by the line joining $P$ with the focus and by the perpendicular from $P$ to the directrix. In addition, the distance of $P$ from the focus is the same as he distance of $P$ from the directrix.

It follows that the reflections of focus $F$ about the coordinate axes: $H=(-2/5,4/5)$ and $K=(2/5,-4/5)$, are also the projections of tangency points on the directrix. Hence line $HK$ is the directrix and the lines from $H$ and $K$, perpendicular to the directrix, intersect y-axis and x-axis at tangency points $A=(0,1)$ and $B=(2,0)$.