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Conic Section

If the curve ...

Question

If the curve $y = a x^{2} + b x$ passes through $(- 1, 0)$ and $y = x$ is the tangent line at $x = 1$ then $(a, b)$ .

(1, 1)

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(1 / 2, 1 / 2)

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(1 / 3, 1 / 3)

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(3, 3)

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Solution

The correct option is C $(1 / 3, 1 / 3)$
As it passes through $(- 1, 0)$ , we have
$0 = a - b$
$a = b$
Also, we have
$\frac{d y}{d x} = 2 a x + b$
$2 a + b = 1$
$3 a = 1$
$a = \frac{1}{3} = b$

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