TITLE: Shortest distance from point to curve
QUESTION [9 upvotes]: I could use some help solving the following problem. I have many more like this but I figured if I learn how to do one then I can figure out the rest on my own. Thanks in advance!

A curve described by the equation $y=\sqrt{16x^2+5x+16}$ on a Cartesian plane. What is the shortest distance between coordinate $(2,0)$ and this line?

REPLY [7 votes]: Start by finding the distance from some point on the curve to $(2,0)$ in terms of $x$. Using the distance formula, we get
$$D=\sqrt{(x-2)^2+(\sqrt{16x^2+5x+16}-0)^2}$$
$$D=\sqrt{x^2-4x+4+16x^2+5x+16}$$
$$D=\sqrt{17x^2+x+20}$$
This will end up being a messy derivative. However, since the distance $D$ will never be negative, we can minimize $D^2$ instead of $D$ and still get the same answer. So now we get
$$D^2=17x^2+x+20$$
$$\frac{dD^2}{dx}=34x+1$$
Now we set this equal to $0$ and solve for $x$:
$$34x+1=0$$
$$x=-\frac{1}{34}$$
So the distance is minimized at $x=-\frac{1}{34}$, and to find the minimum distance, simply evaluate $D$ when $x=-\frac{1}{34}$.