A secant y=kx+b has nonzero k,b and chord midpoint D on the preceding ellipse. Find the slope of OD.

01 / Standard#Your turn

\frac{x^2}8+\frac{y^2}4=1

Official paper · jm01-2025 · II.5(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Eliminate y and apply the root-sum formula.

Hint 2

The midpoint’s ordinate also lies on the secant line.

Worked solution

Substitution gives a quadratic for the two intersection abscissas.
$(1+2k^2)x^2+4kbx+2b^2-8=0\implies x_D=-\frac{2kb}{1+2k^2}$
Recover y_D and divide; k,b nonzero ensure x_D is nonzero.
$y_D=kx_D+b=\frac b{1+2k^2},\quad m_{OD}=\frac{y_D}{x_D}=-\frac1{2k}$

Slope −1/(2k).

Checks and common pitfalls: The midpoint need not lie on the ellipse; it lies on the secant.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Substitution gives a quadratic for the two intersection abscissas.
$(1+2k^2)x^2+4kbx+2b^2-8=0\implies x_D=-\frac{2kb}{1+2k^2}$
Recover y_D and divide; k,b nonzero ensure x_D is nonzero.
$y_D=kx_D+b=\frac b{1+2k^2},\quad m_{OD}=\frac{y_D}{x_D}=-\frac1{2k}$

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

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