For the two roots m₁,m₂ of the slope quadratic, derive their sum and product.

01 / Standard#Your turn

\frac{x^2}{9}+\frac{y^2}{4}=1,\quad A=(h,k)\text{ outside the ellipse}

Official paper · jm02-2023 · 3(a)(ii) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Expand the tangent condition in powers of m.

Hint 2

Apply Vieta with h²≠9.

Worked solution

Form the genuine quadratic assumed in this subpart.
$(9-h^2)m^2+2hkm+4-k^2=0,\quad h^2\ne9$
Read the two symmetric functions.
$m_1+m_2=\frac{2hk}{h^2-9},\quad m_1m_2=\frac{k^2-4}{h^2-9}$

Sum 2hk/(h²−9); product (k²−4)/(h²−9).

Checks and common pitfalls: When h=±3 the slope equation is not quadratic.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Form the genuine quadratic assumed in this subpart.
$(9-h^2)m^2+2hkm+4-k^2=0,\quad h^2\ne9$
Read the two symmetric functions.
$m_1+m_2=\frac{2hk}{h^2-9},\quad m_1m_2=\frac{k^2-4}{h^2-9}$

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

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