Parabola having focus $(1,2)$ touches both axes. Find the equation of directrix.

Question

Parabola having focus $(1,2)$ touches both axes. Find the equation of directrix.
As perpendicular tangents meet at directrix, the directrix passes through origin. So the directrix has equation of the form $y+mx=0$. How do I get the $m$ ?

Answer 1

Hint: The directrix must be perpendicular to the line joining the points (0,0) and (1,2).

Answer 2

Use the property: The reflection of focus about any tangent of the parabola lies on the directrix.

Hence, taking it's reflection along Y axis, we get $(-1,2)$ and taking it's reflection along X axis, we get $(1,-2)$. Since both the points lie on the directrix of the equation, the equation of the directrix is the line joining these two points.

The directrix hence comes out as $y+2x=0$.