Track / Stage 4 / 172 of 340 #432 of 2000

Problem 432

AMC 12 late, AIME early

Combinatorics Difficulty 4.9 Find the answer

3. Given that the ellipse $C_{1}$ and the hyperbola $C_{2}$ share foci $F_{1}(3,0), F_{2}(-3,0)$ , and have coincident minor axes. Then the number of lattice points inside the region enclosed by the intersection points of $C_{1}$ and $C_{2}$ is $\qquad$

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Your answer

Official solution

3.25.

From the problem, we can let
$C_{1}: \frac{x^{2}}{m+9}+\frac{y^{2}}{m}=1, C_{2}: \frac{x^{2}}{9-m}-\frac{y^{2}}{m}=1,$

where $0<m<9$ .
Let the intersection point be $P\left(x_{0}, y_{0}\right)$ . Then, since point $P$ lies on $C_{1}$ and $C_{2}$ , we have $\frac{x_{0}^{2}}{m+9}+\frac{x_{0}^{2}}{9-m}=2$ , which simplifies to $x_{0}^{2}=9-\frac{m^{2}}{9}$ .
Thus, $y_{0}^{2}=\frac{m^{2}}{9}$ , where $x_{0}^{2}, y_{0}^{2} \neq 0,9$ .
Therefore, $x_{0}^{2}+y_{0}^{2}=9$ .
From the sketch (omitted), it is known that there are 25 lattice points within the enclosed region.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.