When m=√5, find the area of triangle AOB.

01 / Standard#Your turn

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

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find |x₁−x₂| from the root sum and product.

Hint 2

Use half the absolute coordinate determinant for area.

Worked solution

Calculate the root separation.
$x_1+x_2=10\sqrt5,\quad x_1x_2=29,\quad|x_1-x_2|=\sqrt{500-116}=8\sqrt6$
Use y=√5x−5 in the determinant.
$[AOB]=\frac12|x_1y_2-x_2y_1|=\frac52|x_2-x_1|=20\sqrt6$

Area 20√6.

Checks and common pitfalls: Area uses the absolute determinant, regardless of vertex ordering.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Calculate the root separation.
$x_1+x_2=10\sqrt5,\quad x_1x_2=29,\quad|x_1-x_2|=\sqrt{500-116}=8\sqrt6$
Use y=√5x−5 in the determinant.
$[AOB]=\frac12|x_1y_2-x_2y_1|=\frac52|x_2-x_1|=20\sqrt6$

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

2024 JM02