Answer
$y=-\frac{1}{4}x+\frac{17}{4}$
Work Step by Step
From geometry, a line tangent to a circle is perpendicular to the radius at the point of tangency. The radius with endpoint $(1,4)$ has slope $m=\frac{0-4}{0-1}=4$, so the slope of the tangent line at $(1, 4)$ is the negative reciprocal of $4$, or $-\frac{1}{4}$. An equation of the tangent line is as follows:
$y-4=-\frac{1}{4}(x-1)\\y=-\frac{1}{4}x+\frac{17}{4}$
