Find the minimum distance between a line and a parabola

Question

Find the minimum distance between the line $y = 4x + 4$ and the parabola $x=y^2$.

Not sure how to solve this problem?

Any help will be appreciated!

Answer 1

It's the distance between the tangent line $yy_1=p(x+x_1)$ to the parabola $y^2=x$ and $y=4x+4$, which parallel to the tangent line.

Since $p=\frac{1}{2}$, we obtain $\frac{1}{2y_1}=4$, which gives $y_1=\frac{1}{8}$ and $\left(\frac{1}{64},\frac{1}{8}\right)$ is a touching point.

Id est, for the distance we obtain: $$\frac{\left|4\cdot\frac{1}{64}-\frac{1}{8}+4\right|}{\sqrt{4^2+(-1)^2}}$$

Answer 2

The minimum distance is at the $x$ at which the tangent to the parabola is $4$. Where the tangent is parallel to the straight line:

So, one has to solve the following equation

$$\frac12x^{-\frac12}=4.$$

The solution is $$x_m=\frac1{64}.$$