Find the point on the graph of $y=e^{2x}$ at which the tangent line passes through the origin.
Completely lost on this question, the wording is confusing here.
[Math] Find the point on the graph of $y=e^{2x}$ at which the tangent line passes through the origin
calculusderivativesexponential function
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Best Answer
We have a mystery point $P$ on the curve. The tangent line at $P$ goes through the origin. We want to find the coordinates of $P$.
Let the coordinates of $P$ be $(a,e^{2a})$. We find the equation of the tangent line to $y=e^{2x}$ at $P$.
The slope of the tangent line at $P$ is $2e^{2a}$. So the equation of the tangent line is $$y-e^{2a} =2e^{2a}(x-a).$$ This line passes through the origin. So $(0,0)$ is on the line, and therefore $$-e^{2a}=(2e^{2a})(-a).$$ Cancel the $e^{2a}$. We get $a=\frac{1}{2}$.