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How to show that a line pass through a point?

Two fixed straight line $OX$ and $OY$ are cut by a variable line at the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the perpendiculars drawn from $A$ and $B$ upon the lines $OBY$ and $OAX$ show that,if $AB$ pass through a fixed point,then $PQ$ will pass through a fixed point.

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    $\begingroup$ The question seems unclear to me. Can you explain what you mean by $AB$ passing through a fixed point? $\endgroup$ – Vishal Gupta Sep 23 '13 at 12:21
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    $\begingroup$ @Vishal: I merged [coordinate-geometry] into [analytic-geometry] (which is a synonym and has existed for a while on MSE). $\endgroup$ – Willie Wong Sep 23 '13 at 12:30
  • $\begingroup$ plz help in solving this.. $\endgroup$ – maths lover Sep 23 '13 at 18:08
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Say $AB$ always passes through $N$. Let $N$ be the point such that $\angle XON=\angle MOY$ and $\displaystyle\frac{ON}{OM}=\cos\theta$, where $\theta=\angle XOY$; we claim that $N$ always lies on line $PQ$. Indeed, since $OQ=OB\cos \theta$ and $OP=OA\cos \theta$, $\triangle OPQ\sim\triangle OAB$ with scale factor $\cos \theta$. Because the corresponding angles $\angle QON$ and $\angle MOB$ are equal, $M$ corresponds to $N$ in $OAB$'s similar triangle, so $N$ lies on $PQ$ always.

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