Show that the nonvertical line through A is tangent exactly when 9m²−(k−mh)²+4=0.

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)(i) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the line as y=mx+d with d=k−mh.

Hint 2

Tangency means a double real intersection root.

Worked solution

Substitute and collect the quadratic in x.
$(4+9m^2)x^2+18mdx+9d^2-36=0$
The leading coefficient is positive, and the discriminant is zero exactly under the required condition.
$\Delta=144(9m^2+4-d^2)=0\iff9m^2-(k-mh)^2+4=0$

The stated condition is necessary and sufficient.

Checks and common pitfalls: This slope form treats only nonvertical tangents; vertical cases require separate handling.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Substitute and collect the quadratic in x.
$(4+9m^2)x^2+18mdx+9d^2-36=0$
The leading coefficient is positive, and the discriminant is zero exactly under the required condition.
$\Delta=144(9m^2+4-d^2)=0\iff9m^2-(k-mh)^2+4=0$

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

2023 JM02