The equation of a curve is $y = 9x^2 + (2m + 1)x + 1 + c$, where $m$ and $c$ are constant

Question

The equation of a curve is $y = 9x^2 + (2m + 1)x + 1 + c$, where $m$ and $c$ are constants. The line $y = mx + c$ is a tangent to the curve at the point $P$. Find the positive value of $m$. Using this value of $m$, and given that the curve passes through $(-2, 19)$, find the coordinates of $P$. The straight line $L$ meets the curve at one point only. Given that $L$ is not a tangent to the curve, describe all possible equations of $L$.

Accepted Answer

(i) $m=5$ (ii) $P\left(-\dfrac13,\dfrac73\right)$ (iii) No unique line; every vertical line $x=\lambda$ meets the curve once and is not a tangent

Differentiation - Tangents and Normals at a Point

Differentiation - Tangents and Normals at a Point

Solution:

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