TITLE: If $x$ is a nonnegative real number, find the minimum value of $\sqrt{x^2 +4} + \sqrt{x^2 -24x+153}$
QUESTION [6 upvotes]: If $x$ is a nonnegative real number, then find the minimum value of
$$\sqrt{x^2 +4} + \sqrt{x^2 -24x+153}$$
How can I approach this? Thanks

REPLY [8 votes]: Thinking geometrically, you want to minimise the sum of distances from $(x,0)$ to $(0,2)$ and $(x,0)$ to $(12,3)$. Reflect the point $(12,3)$ in the $x$-axis.
The total length of the two line segments from $(0,2)$ to $(x,0)$ and $(x,0)$ to $(12,-3)$ will be smallest when they form a straight line.
Hence the minimum sum of distances is the distance between $(0,2)$ and $(12,-3)$, which is $13$.