An Asymptote Triangle

★ 8/10Monotonicity conic sections AM-GM analytic geometry

Problem

Let $P$ be a point of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$ ), and let $A, B$ move along the two asymptotes of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ , with $\overrightarrow{AP} = \lambda\overrightarrow{PB}$ for a constant $\lambda$ . Let $O$ be the origin. If the maximum area of $\triangle AOB$ equals

\frac{a^2 + b^2}{a + b}\cdot\frac{(1+\lambda)^2}{4|\lambda|},

find the range of possible values of $\dfrac{1}{a} + \dfrac{7}{b}$ .

Answer

Solution

Difficulty	8/10
Topics	Monotonicity, conic sections, AM-GM, analytic geometry

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments

No comments yet.