The line y=x+m meets this hyperbola at P,Q, and OP⊥OQ. Find m.

01 / Standard#Your turn

\frac{x^2}{2}-\frac{y^2}{3}=1

Official paper · jm01-2024 · II.5(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use Vieta instead of solving for both intersections.

Hint 2

Perpendicular position vectors have zero dot product.

Worked solution

Substitute the line and read the root sum/product.
$x^2-4mx-2m^2-6=0,\quad x_1+x_2=4m,\quad x_1x_2=-2m^2-6$
Impose the dot product condition.
$0=x_1x_2+(x_1+m)(x_2+m)=2(-2m^2-6)+4m^2+m^2=m^2-12$
Both signs give two distinct intersections.
$m=\pm2\sqrt3,\quad\Delta=24m^2+24>0$

m=±2√3.

Checks and common pitfalls: A perpendicularity condition uses a dot product, not equal slopes.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Substitute the line and read the root sum/product.
$x^2-4mx-2m^2-6=0,\quad x_1+x_2=4m,\quad x_1x_2=-2m^2-6$
Impose the dot product condition.
$0=x_1x_2+(x_1+m)(x_2+m)=2(-2m^2-6)+4m^2+m^2=m^2-12$
Both signs give two distinct intersections.
$m=\pm2\sqrt3,\quad\Delta=24m^2+24>0$

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

2024 JM01