Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.3 AIME Find the answer China

Assume that $M = \{(x, y) \mid x^2 + 2y^2 = 3\}$ , and $N = \{(x, y) \mid y = mx + b\}$ . If $M \cap N \neq \emptyset$ for all $m \in \mathbb{R}$ , then $b$ takes values from:

Pick one

Solution

For any $m \in \mathbb{R}$ we have $M \cap N \neq \emptyset$ , which means point $(0, b)$ is on or in the ellipsoid $\frac{x^2}{3} + \frac{2y^2}{3} = 1$ . Therefore
$\frac{2b^2}{3} \le 1, \text{ or } -\frac{\sqrt{6}}{2} \le b \le \frac{\sqrt{6}}{2}.$
Answer: A.

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