Track / Stage 3 / 89 of 260 #729 of 2604

Problem 729

AMC 10/12, early questions

Geometry Difficulty 3.5 Find the answer China Mathematical Competition · China

In ellipse $\Gamma$ , $A$ is an endpoint of the major axis, $B$ is an endpoint of the minor axis, and $F_1, F_2$ are the foci. If $\overrightarrow{AF_1} \cdot \overrightarrow{AF_2} + \overrightarrow{BF_1} \cdot \overrightarrow{BF_2} = 0$ , then the value of $\frac{|AB|}{|F_1F_2|}$ is ______.

Your answer

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

Official solution

Without loss of generality, suppose the equation of $\Gamma$ is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ( $a > b > 0$ ), and $A(a, 0)$ , $B(0, b)$ , $F_1(-c, 0)$ , $F_2(c, 0)$ . By the given conditions, we get
$\overrightarrow{AF_1} \cdot \overrightarrow{AF_2} + \overrightarrow{BF_1} \cdot \overrightarrow{BF_2} = (-c-a)(c-a) + (-c^2+b^2) = a^2+b^2-2c^2 = 0.$
Therefore, $\frac{|AB|}{|F_1F_2|} = \frac{\sqrt{a^2+b^2}}{2c} = \frac{\sqrt{2c^2}}{2c} = \frac{\sqrt{2}}{2}$ . $\square$

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.