Let O be the vertex of the parabola y^2=4x, and let its chords OP and OQ be perpendicular …

Question

Let $O$ be the vertex of the parabola $y^2=4x$, and let its chords $OP$ and $OQ$ be perpendicular to each other. If the locus of the midpoint of the line segment $PQ$ is a conic $C$, then the length of its latus rectum is:

Accepted Answer

Here $a=1$; for perpendicular chords through the vertex, $t_1t_2=-4$. With $P=(t_1^2,2t_1),\ Q=(t_2^2,2t_2)$, midpoint $(h,k)$:

$$2h=t_1^2+t_2^2=(t_1+t_2)^2-2t_1t_2 = \left(\frac{2k}{2}\right)^2+8=k^2+8$$

(using $2k=2(t_1+t_2)\Rightarrow t_1+t_2=k$), so $k^2=2h-8$, i.e. $y^2=2(x-4)$. Here $4A=2\Rightarrow$ latus rectum $=2$.

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