Find the equations of the tangents to the curve $f(x) = 2x^2 + 3$ that pass through the point $(2,3)$. Include a sketch as part of your solution.
First, I found the derivative of $f(x)$ which is $4x$, but I'm not sure what to do next. Thank you.
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Any point on the curve $y=f(x)$ is of the form $\langle a,2a^2+3\rangle$, and the slope of the tangent line at that point is, as you discovered, $4a$. Write down the equation of the straight line of slope $4a$ passing through the point $\langle a,2a^2+3\rangle$, then choose $a$ so that this line also passes through the point $\langle 2,3\rangle$.
Brian M. Scott
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Would the equation of the straight line of slope 4a passing through the point (a, 2a^2 + 3) be 2a^2 + 3 = (4a)a? – user148748 Oct 27 '14 at 02:32
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@user148748: That can’t be: its equation will have the form $y=mx+b$ for some constants $m$ and $b$ (which will depend on $a$). – Brian M. Scott Oct 27 '14 at 02:37
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Observe that a straight $g$ through $(2,3)$ has the form $g(x)=mx +3-2m$. Now determine all $m$ for which $f(x)=g(x)$ possesses exactly one solution.
Michael Hoppe
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