Consider every tangent line to the curve y = x³ that passes through the point (2, 0). Find the sum of their slopes. Give the answer as an integer.

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Tangents from an outside point

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Consider every tangent line to the curve $y = x^{3}$ that passes through the point $(2, 0)$ . Find the sum of their slopes. Give the answer as an integer.

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Hint

Write the tangent at a general point $(a, a^{3})$ and ask when it hits $(2, 0)$ .

Solution

The tangent at $(a, a^{3})$ has slope $3 a^{2}$ , so its equation is $y = a^{3} + 3 a^{2} (x - a) = 3 a^{2} x - 2 a^{3}$ .
It passes through $(2, 0)$ when $6 a^{2} - 2 a^{3} = 0$ , that is $2 a^{2} (3 - a) = 0$ .
So $a = 0$ or $a = 3$ , giving the slopes $0$ and $3 \times 9 = 27$ .
The sum of the slopes is $0 + 27 = 27$ .

Answer: $27$

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