An Interior Point Minimum

★ 9/10AM-GM analytic geometry inequality Vieta's Formulas Rotation

Problem

Let $M(x_0, y_0)$ be a point in the first quadrant interior to the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ( $a, b > 0$ ). Lines through $M$ meet the right branch at $A$ and $B$ ; suppose the minimum area of $\triangle AOB$ (with $O$ the origin) equals

\sqrt{b^2x_0^2 - a^2y_0^2}.

Find the minimum value of $\dfrac{3x_0}{a} - \dfrac{y_0}{b}$ .

Answer

Solution

Difficulty	9/10
Topics	AM-GM, analytic geometry, inequality, Vieta's Formulas, Rotation

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