A=(−1,0), B=(1,0). Prove the locus condition describes y²=4x, including the converse.

01 / Standard#Your turn

\overrightarrow{AN}\cdot\overrightarrow{AB}=|\overrightarrow{BN}|\,|\overrightarrow{AB}|,\quad N=(x,y)

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Compute each vector and its length.

Hint 2

Record the nonnegative side before squaring.

Worked solution

Write the dot-product equation in coordinates.
$2(x+1)=2\sqrt{(x-1)^2+y^2}\implies x+1\ge0$
Square and simplify.
$(x+1)^2=(x-1)^2+y^2\iff y^2=4x$
Conversely y²=4x forces x≥0, so x+1>0 and reversing the square is valid.

Exactly the parabola y²=4x.

Checks and common pitfalls: Squaring a locus equation requires a converse/sign check.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Write the dot-product equation in coordinates.
$2(x+1)=2\sqrt{(x-1)^2+y^2}\implies x+1\ge0$
Square and simplify.
$(x+1)^2=(x-1)^2+y^2\iff y^2=4x$
Conversely y²=4x forces x≥0, so x+1>0 and reversing the square is valid.

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

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