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?
Best Answer
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 $x=-\frac{1}{34}$.