Derive the completed-square identity from the stated relation.

01 / Standard#Your turn

x,y>0,\quad0<m<1,\quad y^2-2mxy+x^2=a^2

Official paper · jm01-2021 · II.3(a) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Complete the square in x, keeping y as a parameter.

Hint 2

The cross term must remain −2mxy.

Worked solution

Split the y² coefficient after completing the x square.
$x^2-2mxy+y^2=(x-my)^2+(1-m^2)y^2$
Use the original equality and rearrange.
$(1-m^2)y^2=a^2-(x-my)^2$

(1−m²)y²=a²−(x−my)².

Checks and common pitfalls: The square is x−my, not y−mx in this particular rearrangement.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Split the y² coefficient after completing the x square.
$x^2-2mxy+y^2=(x-my)^2+(1-m^2)y^2$
Use the original equality and rearrange.
$(1-m^2)y^2=a^2-(x-my)^2$

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

2021 JM01