Find all real slopes m giving two distinct intersections.

01 / Standard#Your turn

H:x^2-\frac{y^2}{4}=1,\quad L:y=m(x-\sqrt5)

Official paper · jm02-2024 · 3(b) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A genuine quadratic requires m²−4≠0.

Hint 2

Then test a strictly positive discriminant.

Worked solution

Calculate the discriminant.
$\Delta=20m^4-4(m^2-4)(5m^2+4)=64(m^2+1)>0$
Only the degenerate leading-coefficient values are excluded.
$m\in\mathbb R\setminus\{-2,2\}$

All real m except ±2.

Checks and common pitfalls: A positive discriminant alone is insufficient when the equation becomes linear.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Calculate the discriminant.
$\Delta=20m^4-4(m^2-4)(5m^2+4)=64(m^2+1)>0$
Only the degenerate leading-coefficient values are excluded.
$m\in\mathbb R\setminus\{-2,2\}$

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

2024 JM02