If the two tangents in part (b) are perpendicular, find the complete locus of A.

01 / Standard#Your turn

P:x^2=8y

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Both slopes are finite, so m₁m₂=−1.

Hint 2

Show every real h occurs, not merely that k=−2.

Worked solution

The ordinate is fixed by the slope product.
$k=2m_1m_2=-2$
For any real h, solve the tangent-slope equation.
$2m^2-hm-2=0,\quad\Delta=h^2+16>0$
Its two distinct real roots have product −1, so their tangents meet at (h,−2) and are perpendicular.
$m_1+m_2=h/2,\quad m_1m_2=-1$

The full straight line y=−2.

Checks and common pitfalls: A necessary equation for a locus also needs a sufficiency argument.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The ordinate is fixed by the slope product.
$k=2m_1m_2=-2$
For any real h, solve the tangent-slope equation.
$2m^2-hm-2=0,\quad\Delta=h^2+16>0$
Its two distinct real roots have product −1, so their tangents meet at (h,−2) and are perpendicular.
$m_1+m_2=h/2,\quad m_1m_2=-1$

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

2021 JM02