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Problem 729

AMC 10/12, early questions

Geometry Difficulty 3.5 Find the answer China Mathematical Competition · China

In the plane rectangular coordinate system, given hyperbola $\Gamma: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ( $a, b > 0$ ), a line with inclination angle $\frac{\pi}{4}$ passes through a vertex of $\Gamma$ and another point $(2, 3)$ on it. Then the eccentricity of $\Gamma$ is ______.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

The slope of the line described in the question is $1$ and it passes through point $(2, 3)$ , so its equation is $y = x + 1$ . This line intersects with the $x$ -axis at point $(-1, 0)$ , and thus $(-1, 0)$ is a vertex of $\Gamma$ . Hence, $a = 1$ . And since point $(2, 3)$ is on $\Gamma$ , we know that $\frac{2^2}{1} - \frac{3^2}{b^2} = 1$ , and then $b^2 = 3$ . Denote $c = \sqrt{a^2 + b^2}$ . Then the eccentricity of $\Gamma$ is $\frac{c}{a} = \frac{\sqrt{a^2 + b^2}}{a} = 2$ .

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