Find tan∠QMF₁ for the preceding points.

01 / Standard#Your turn

F_1=(-\sqrt5,0),\quad M=\left(\frac{6\sqrt5}5,-\frac{2\sqrt5}5\right)

Official paper · jm02-2026 · 3(c) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the slopes of MQ and MF₁.

Hint 2

Check that the angle between the two rays is acute.

Worked solution

Find the two slopes and check the ray dot product is positive.
$m_{MQ}=-2,\quad m_{MF_1}=-\frac2{11},\quad\overrightarrow{MQ}\cdot\overrightarrow{MF_1}=9>0$
Apply the tangent of the angle between lines.
$\tan\angle QMF_1=\left|\frac{-2+2/11}{1+4/11}\right|=\frac43$

tan∠QMF₁=4/3.

Checks and common pitfalls: An absolute-value slope formula gives an acute line angle; check the requested ray angle.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Find the two slopes and check the ray dot product is positive.
$m_{MQ}=-2,\quad m_{MF_1}=-\frac2{11},\quad\overrightarrow{MQ}\cdot\overrightarrow{MF_1}=9>0$
Apply the tangent of the angle between lines.
$\tan\angle QMF_1=\left|\frac{-2+2/11}{1+4/11}\right|=\frac43$

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

2026 JM02