For some a > 0, the tangent line to y = x² at the point (a, a²) passes through the point (1, −3). Find a. Give the answer as an integer.

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A tangent through a point

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For some $a > 0$ , the tangent line to $y = x^{2}$ at the point $(a, a^{2})$ passes through the point $(1, - 3)$ . Find $a$ . Give the answer as an integer.

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Hint

Write the tangent line in terms of $a$ and require it to pass through the given point.

Solution

The slope at $x = a$ is $2 a$ , so the tangent is $y = a^{2} + 2 a (x - a) = 2 a x - a^{2}$ .
Requiring the point $(1, - 3)$ gives $2 a - a^{2} = - 3$ , that is $a^{2} - 2 a - 3 = 0$ .
This factors as $(a - 3) (a + 1) = 0$ , so $a = 3$ or $a = - 1$ .
Only the positive root qualifies, so $a = 3$ .

Answer: $3$

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