Two distinct tangents to x²=8y have slopes m₁,m₂ and meet at A(h,k). Find A in terms of the slopes.

01 / Standard#Your turn

P:x^2=8y

Official paper · jm02-2021 · 3(b)(i) · PDF 5

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use y=mᵢx−2mᵢ² from part (a).

Hint 2

Subtract the two incidence equations; m₁≠m₂.

Worked solution

The two lines have different slopes, since a slope specifies one tangent of this parabola.
$k=m_1h-2m_1^2=m_2h-2m_2^2$
Cancel the nonzero slope difference.
$(m_1-m_2)h=2(m_1-m_2)(m_1+m_2)\implies h=2(m_1+m_2)$
Substitute back to obtain the ordinate.
$k=2m_1(m_1+m_2)-2m_1^2=2m_1m_2$

A=(2(m₁+m₂),2m₁m₂).

Checks and common pitfalls: Cancellation requires the stated distinctness of the tangents.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The two lines have different slopes, since a slope specifies one tangent of this parabola.
$k=m_1h-2m_1^2=m_2h-2m_2^2$
Cancel the nonzero slope difference.
$(m_1-m_2)h=2(m_1-m_2)(m_1+m_2)\implies h=2(m_1+m_2)$
Substitute back to obtain the ordinate.
$k=2m_1(m_1+m_2)-2m_1^2=2m_1m_2$

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

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