A slope-1 line meets the parabola at C,D and the y-axis at M, with M between C,D as in the diagram. If DM=3CM, find CD.

01 / Standard#Your turn

x^2=4y

Official paper · jm01-2023 · II.2(c) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Let the line be y=x+t.

Hint 2

Opposite sides of M give opposite-signed abscissae.

Worked solution

Vieta gives the sum, and the diagram gives the signed ratio.
$x^2-4x-4t=0,\quad x_C+x_D=4,\quad x_D=-3x_C$
Solve and convert the horizontal span to chord length.
$x_C=-2,\ x_D=6,\quad CD=\sqrt{1+1^2}(6+2)=8\sqrt2$

CD=8√2.

Checks and common pitfalls: The diagram’s betweenness condition is essential; unsigned lengths alone allow another configuration.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Vieta gives the sum, and the diagram gives the signed ratio.
$x^2-4x-4t=0,\quad x_C+x_D=4,\quad x_D=-3x_C$
Solve and convert the horizontal span to chord length.
$x_C=-2,\ x_D=6,\quad CD=\sqrt{1+1^2}(6+2)=8\sqrt2$

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

2023 JM01