At A=(5,4), find the acute angle between the tangents, expressed using arctan.

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(c) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the root sum and product without solving each slope.

Hint 2

The slope difference squared is the sum squared minus four times the product.

Worked solution

Evaluate the symmetric quantities.
$m_1+m_2=5/2,\quad m_1m_2=3/4,\quad|m_1-m_2|=\sqrt{13}/2$
Use the acute line-angle formula.
$\tan\alpha=\frac{|m_1-m_2|}{|1+m_1m_2|}=\frac{2\sqrt{13}}7\implies\alpha=\arctan\frac{2\sqrt{13}}7$

α=arctan(2√13/7).

Checks and common pitfalls: Use the absolute ratio to obtain the angle between unoriented lines.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Evaluate the symmetric quantities.
$m_1+m_2=5/2,\quad m_1m_2=3/4,\quad|m_1-m_2|=\sqrt{13}/2$
Use the acute line-angle formula.
$\tan\alpha=\frac{|m_1-m_2|}{|1+m_1m_2|}=\frac{2\sqrt{13}}7\implies\alpha=\arctan\frac{2\sqrt{13}}7$

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

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