Let P be a parabola with vertex (2,3) and directrix 2x+y=6.

Question

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E$ (equation not preserved) of eccentricity $\dfrac{1}{\sqrt2}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$ is:

Accepted Answer

Foot of perpendicular from the vertex $(2,3)$ to $2x+y=6$: $(2,3)-\dfrac{(4+3-6)}{5}(2,1)=\left(\dfrac85,\dfrac{14}{5}\right)$.

Focus $=2(2,3)-\left(\dfrac85,\dfrac{14}{5}\right)=\left(\dfrac{12}{5},\dfrac{16}{5}\right)$.

*(The equation of $E$ was not preserved in the source, so the final latus-rectum computation cannot be independently verified. The stated answer is $656/25$.)*

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