The hyperbola has eccentricity 3. Minimise (b²+2)/a for a,b>0.

01 / Standard#Your turn

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

Official paper · jm01-2023 · I.12 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Convert eccentricity into a relation between a and b.

Hint 2

Apply AM–GM to two positive terms.

Worked solution

Use c²=a²+b² and c/a=3.
$b^2=8a^2,\quad\frac{b^2+2}{a}=8a+\frac2a$
The product is constant and equality is attainable.
$8a+2/a\ge2\sqrt{16}=8,\quad8a=2/a\iff a=1/2$

E: 8.

Checks and common pitfalls: The numerator is b²+2 as a whole.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Use c²=a²+b² and c/a=3.
$b^2=8a^2,\quad\frac{b^2+2}{a}=8a+\frac2a$
The product is constant and equality is attainable.
$8a+2/a\ge2\sqrt{16}=8,\quad8a=2/a\iff a=1/2$

Think first. Reveal a hint when the class is ready.

2023 JM01