Find the other intersection M of QF₂ with the ellipse.

01 / Standard#Your turn

Q=\left(\frac{3\sqrt5}5,\frac{4\sqrt5}5\right),\quad F_2=(\sqrt5,0)

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find the slope and equation of QF₂.

Hint 2

One of the quadratic intersection roots already corresponds to Q.

Worked solution

Compute the line.
$m=-2,\quad y=-2x+2\sqrt5$
Substitute into the ellipse, then select the other root.
$5x^2-9\sqrt5x+18=0,\quad x_Q+x_M=\frac{9\sqrt5}5\implies x_M=\frac{6\sqrt5}5$
Recover the ordinate from the line.
$y_M=-\frac{2\sqrt5}5$

M=(6√5/5,−2√5/5).

Checks and common pitfalls: Returning Q again does not satisfy the requested distinct intersection.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Compute the line.
$m=-2,\quad y=-2x+2\sqrt5$
Substitute into the ellipse, then select the other root.
$5x^2-9\sqrt5x+18=0,\quad x_Q+x_M=\frac{9\sqrt5}5\implies x_M=\frac{6\sqrt5}5$
Recover the ordinate from the line.
$y_M=-\frac{2\sqrt5}5$

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

2026 JM02