A nonvertical line y=ax+b is tangent to y²=4x. Prove ab=1.

01 / Standard#Your turn

Official paper · jm02-2022 · 3(b) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Parameterise the contact point as (t²/4,t).

Hint 2

A nonvertical tangent has t≠0 and slope 2/t.

Worked solution

Differentiate implicitly and use the tangent point.
$2y\,y'=4\implies a=2/t,\quad t\ne0$
The tangent intercept is t−a·t²/4=t/2.
$b=t/2\implies ab=(2/t)(t/2)=1$

ab=1.

Checks and common pitfalls: The vertex tangent x=0 is vertical and is outside the given line form.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Differentiate implicitly and use the tangent point.
$2y\,y'=4\implies a=2/t,\quad t\ne0$
The tangent intercept is t−a·t²/4=t/2.
$b=t/2\implies ab=(2/t)(t/2)=1$

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

2022 JM02