The angle between the two tangents is π/4 and m₁=2. Find all possible A.

01 / Standard#Your turn

P:x^2=8y

Official paper · jm02-2021 · 3(b)(iii) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the absolute tangent of the angle between the lines.

Hint 2

Retain both solutions after squaring and check the denominator.

Worked solution

Perpendicular slopes are excluded by the specified π/4 angle.
$1=\left|\frac{2-m_2}{1+2m_2}\right|,\quad m_2\ne-1/2$
Squaring and factoring gives two valid slopes.
$(2-m_2)^2=(1+2m_2)^2\iff3m_2^2+8m_2-3=0\implies m_2=-3,1/3$
Use the sum and product formulas from part (i).
$A=(-2,-12)\quad\text{or}\quad A=(14/3,4/3)$

A=(−2,−12) or (14/3,4/3).

Checks and common pitfalls: Using a signed angle formula without the absolute value would lose one of the two unoriented-line configurations.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Perpendicular slopes are excluded by the specified π/4 angle.
$1=\left|\frac{2-m_2}{1+2m_2}\right|,\quad m_2\ne-1/2$
Squaring and factoring gives two valid slopes.
$(2-m_2)^2=(1+2m_2)^2\iff3m_2^2+8m_2-3=0\implies m_2=-3,1/3$
Use the sum and product formulas from part (i).
$A=(-2,-12)\quad\text{or}\quad A=(14/3,4/3)$

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

2021 JM02