If y=mx+c is tangent to x²=8y, prove c=−2m².

01 / Standard#Your turn

P:x^2=8y

Official paper · jm02-2021 · 3(a) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Substitute the line into the parabola.

Hint 2

The intersection quadratic has a repeated root at tangency.

Worked solution

The leading quadratic coefficient remains one for every m.
$x^2-8mx-8c=0$
Set its discriminant to zero and rearrange.
$\Delta=64m^2+32c=0\implies c=-2m^2$

c=−2m².

Checks and common pitfalls: Here the elimination remains quadratic, so the double-root criterion is valid even when m=0.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The leading quadratic coefficient remains one for every m.
$x^2-8mx-8c=0$
Set its discriminant to zero and rearrange.
$\Delta=64m^2+32c=0\implies c=-2m^2$

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

2021 JM02