Let P_1 be a parabola with vertex (3,2) and focus (4,4), and P_2 be its mirror image with …

Question

Let $P_1$ be a parabola with vertex $(3,2)$ and focus $(4,4)$, and $P_2$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of $P_2$ is $x+2y=$

Answer: 10

Accepted Answer

$P_1$'s axis direction is $(1,2)/\sqrt5$ ($a=\sqrt5$), so its directrix passes through $(3,2)-\sqrt5\cdot\frac{(1,2)}{\sqrt5}=(2,0)$ and is $x+2y=2$.

This directrix is parallel to the mirror line $x+2y=6$; reflecting the constant across it:
$$x+2y = 2\cdot6-2=10.$$

Solutions · 1