A Tangent Line, Two Asymptotes, and a Circle Through the Foci

★ 7/10plane geometry Inscribed Angle Similar Triangles conic sections analytic geometry

Problem

As shown in the figure, $F_1$ and $F_2$ are the two foci of the hyperbola

x^2 - \dfrac{y^2}{4} = 1,

and $O$ is the origin. A line is tangent to the right branch of the hyperbola and meets the two asymptotes at points $A$ and $B$ .

1. Prove that $|OA| \cdot |OB| = |OF_1|^2$ . 2. Prove that the four points $F_1$ , $F_2$ , $A$ , $B$ lie on one circle.

Answer

Solution

Difficulty	7/10
Topics	plane geometry, Inscribed Angle, Similar Triangles, conic sections, analytic geometry

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