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Through the vertex O of a parabola y2=4ax, chords OP & OQ are drawn at right angles to one another. If PQ cuts the axis of the parabola at R, find distance OR.

The answer given is that OR equals latus rectum of the parabola. I approached it this way: From the fact that products of slopes of OP and OQ give -1, assuming P $(at_1^2,2at_1)$ and Q $(at_2^2,2at_2)$ I got $t_1t_2=-4$. I used the following equation, putting y=0 to find x: $$ (y-2at_1) = 2a(t_2-t_1)(x-at_1^2)/a(t_2^2-t_1^2)) $$. I got $x = -at_1^2-at_1t_2+at_1^2$. Have I done something wrong or is there something more to be done? Was my approach correct?

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You found $t_1 t_2 = -4,$ and therefore \begin{align} -at_1^2-at_1t_2+at_1^2 &= -at_1t_2 +at_1^2-at_1^2 \\ &= -at_1t_2 +(at_1^2-at_1^2) \\ &= -at_1t_2 \\ &= -a(-4) \\ &= 4a \\ \end{align} which is the latus rectum.

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