Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.5 Junior Find the answer

（This question is worth 10 points）

For what value(s) of $m$ does the line $L: y = x + m$ tangent to, intersect with, or not intersect (disjoint from) the ellipse $9x^2 + 16y^2 = 144$ ?

Solution

Answer:

Solution: Substituting $y = x + m$ into $9x^2 + 16y^2 = 144$ , we get

$9x^2 + 16(x + m)^2 = 144$ .

Simplifying, we obtain $25x^2 + 32mx + 16m^2 - 144 = 0$ .

Since $\Delta = (32m)^2 - 4 \cdot 25 \cdot (16m^2 - 144) = -576m^2 + 14400$ ,

- When $\Delta > 0$ , i.e., $-5 5$ or $m 5 \text{ or } m < -5}$ for being disjoint.

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