Analytic Geometry – Problem 19: Find

Consider the ellipse $C:\frac{x^2}{9}+\frac{y^2}{n}=1\,(0<n<9)$. Point $M$ does not coincide with either focus of $C$. Let $A$ and $B$ be the reflections of $M$ about the two foci of $C$, respectively. Let $N$ be a point such that $Q$, the midpoint of segment $MN$, lies on the ellipse. Find $|AN|+|BN|$.

A. 6 B. 8 C. 12 D. 36

Let the foci of $C$ be $F_1,F_2$. Since $A$ is the reflection of $M$ about $F_1$, $F_1$ is the midpoint of $MA$. Since $B$ is the reflection of $M$ about $F_2$, $F_2$ is the midpoint of $MB$.

Because $Q$ is the midpoint of $MN$, in triangle $MAN$ we have $$|QF_1|=\frac12|AN|.$$ Similarly, in triangle $MBN$, $$|QF_2|=\frac12|BN|.$$ Hence $$|AN|+|BN|=2\bigl(|QF_1|+|QF_2|\bigr).$$

Since $Q\in C$, by the ellipse definition $|QF_1|+|QF_2|=2a$. From $\frac{x^2}{9}+\frac{y^2}{n}=1$ we have $a^2=9\Rightarrow a=3$, so $2a=6$. Therefore $$|AN|+|BN|=2\cdot 6=12.$$ So the correct choice is C.

$|AN|+|BN|=12$ (option C).