Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.3 AIME, harder Find the answer

7. For the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ , the right focus is $F$ . The line $l: y=k x+d$ does not pass through point $F$ , and intersects the right branch of the hyperbola at points $P$ and $Q$ . If the external angle bisector of $\angle P F Q$ intersects $l$ at point $A$ , then the x-coordinate of point $A$ is $\qquad$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

7. $\frac{a^{2}}{\sqrt{a^{2}+b^{2}}}$ .

Draw a perpendicular line $h$ from point $A$ to the $x$ -axis.
Let the projections of $P$ and $Q$ on line $h$ be $P^{\prime}$ and $Q^{\prime}$ , respectively. Then $\frac{P P^{\prime}}{Q Q^{\prime}}=\frac{A P}{A Q}=\frac{F P}{F Q}$ , which means $\frac{P F}{P P^{\prime}}=\frac{Q F}{Q Q^{\prime}}$ . Therefore, line $h$ is the right directrix of the hyperbola. Hence, the x-coordinate of point $A$ is $\frac{a^{2}}{\sqrt{a^{2}+b^{2}}}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.