Math CentralQuandaries & Queries


Question from Aquilah, a student:

For the curve y = x2 + 3x, find the equations of all tangent lines for this graph that also go through the point (3, 14).


Suppose the point $(a, b)$ is on the graph of $y = x^2 + 3x$ and the tangent to the curve at this point passes through $(3, 14).$ Since $(a, b)$ is on the curve $b = a^2 + 3a$ and hence $(a, b) = (a, a^2 + 3a).$

What is the slope of the line joining $(a, a^2 + 3a)$ and $(3,14)?$ Using the derivative what is the slope of the tangent to $y = x^2 + 3x$ at $(a,b)?$ These slopes are equal. Solve for $a.$


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