A secant y=k₁x+d, with k₁d≠0, meets that ellipse at A,B. M is the midpoint, and OM has slope k₂. Prove k₁k₂=−2.

01 / Standard#Your turn

x^2/16+y^2/32=1

Official paper · jm01-2022 · II.3(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use Vieta for the intersection abscissae.

Hint 2

The midpoint also lies on the secant.

Worked solution

Obtain the midpoint abscissa.
$(2+k_1^2)x^2+2k_1dx+d^2-32=0\implies x_M=-\frac{k_1d}{2+k_1^2}$
Compute its ordinate and ratio; xM≠0 follows from k₁d≠0.
$y_M=\frac{2d}{2+k_1^2},\quad k_2=\frac{y_M}{x_M}=-\frac2{k_1}\implies k_1k_2=-2$

k₁k₂=−2.

Checks and common pitfalls: The nonzero slope and intercept conditions ensure OM has a defined finite slope.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Obtain the midpoint abscissa.
$(2+k_1^2)x^2+2k_1dx+d^2-32=0\implies x_M=-\frac{k_1d}{2+k_1^2}$
Compute its ordinate and ratio; xM≠0 follows from k₁d≠0.
$y_M=\frac{2d}{2+k_1^2},\quad k_2=\frac{y_M}{x_M}=-\frac2{k_1}\implies k_1k_2=-2$

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

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