Find slopes for which OA⊥OB, where A,B are the intersections and O is the origin.

01 / Standard#Your turn

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

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the root sum and product from part (a).

Hint 2

Set the position-vector dot product equal to zero.

Worked solution

Read Vieta’s formulae.
$s=x_1+x_2=\frac{2\sqrt5m^2}{m^2-4},\quad p=x_1x_2=\frac{5m^2+4}{m^2-4}$
Expand the dot product and substitute.
$0=p+m^2(p-\sqrt5s+5)=\frac{4-11m^2}{m^2-4}$
Solve and check the excluded values.
$m=\pm\frac2{\sqrt{11}}=\pm\frac{2\sqrt{11}}{11}\ne\pm2$

m=±2√11/11.

Checks and common pitfalls: The denominator restriction m≠±2 remains in force.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Read Vieta’s formulae.
$s=x_1+x_2=\frac{2\sqrt5m^2}{m^2-4},\quad p=x_1x_2=\frac{5m^2+4}{m^2-4}$
Expand the dot product and substitute.
$0=p+m^2(p-\sqrt5s+5)=\frac{4-11m^2}{m^2-4}$
Solve and check the excluded values.
$m=\pm\frac2{\sqrt{11}}=\pm\frac{2\sqrt{11}}{11}\ne\pm2$

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

2024 JM02