Find both intercepts m for a tangent parallel to y=x−1.

01 / Standard#Your turn

y=x+m,\quad\frac{x^2}2+y^2=1

Official paper · jm02-2025 · 3(d) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Substitute the line into the ellipse.

Hint 2

Tangency means a repeated intersection root.

Worked solution

Obtain the quadratic in x.
$3x^2+4mx+2m^2-2=0$
Set its discriminant to zero.
$16m^2-12(2m^2-2)=0\implies m^2=3\implies m=\pm\sqrt3$

m=±√3.

Checks and common pitfalls: There are two parallel tangents, on opposite sides of the ellipse.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Obtain the quadratic in x.
$3x^2+4mx+2m^2-2=0$
Set its discriminant to zero.
$16m^2-12(2m^2-2)=0\implies m^2=3\implies m=\pm\sqrt3$

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

2025 JM02