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Problem 229

AMC 10/12, early questions

Geometry Difficulty 3.8 Find the answer

Consider the hyperbola $M$ : ${x^{2}} - \frac{{y^{2}}}{{{b^{2}}}} = 1$ . Its left and right foci are $F_{1}$ and $F_{2}$ , respectively, with $|F_{1}F_{2}|=2c$ . A circle with center at the origin $O$ and radius $c$ intersects with the hyperbola $M$ at point $P$ in the first quadrant. If $|PF_{1}|=c+2$ , find the x-coordinate of point $P$ .

Your answer

Official solution

From the given information, we know that the real axis of the hyperbola is $2a=2$ . Since $P$ is a point on the hyperbola in the first quadrant, we have $|PF_{1}|-|PF_{2}|=2a=2$ . Given $|PF_{1}|=c+2$ , we have $|PF_{2}|=c$ . Consequently, $\triangle OPF_{2}$ is an equilateral triangle with side length $c$ , which implies that the coordinates of point $P$ are $(\frac{c}{2}, \frac{\sqrt{3}c}{2})$ .

Substitute these coordinates into the hyperbola's equation:
$\left(\frac{c}{2}\right)^{2} - \frac{\left(\frac{\sqrt{3}c}{2}\right)^{2}}{b^{2}} = 1.$

Since $c^{2} = a^{2} + b^{2} = 1 + b^{2}$ , solve for $c$ to obtain $c=\sqrt{3}+1$ . Therefore, the x-coordinate of point $P$ is $\frac{c}{2} = \boxed{\frac{\sqrt{3}+1}{2}}$ .

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