If $x$ is a nonnegative real number, find the minimum value of $\sqrt{x^2 +4} + \sqrt{x^2 -24x+153}$

Question

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

Answer 1

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 $13$.

Answer 2

Process 1: Derivate it to zero

Process 2: Using triangle inequality,

Given,$$\sqrt{x^2 +4} + \sqrt{x^2 -24x+153}$$

$$\rightarrow\sqrt{x^2+2^2}+\sqrt{\left(-x+12\right)^2+3^2}\geq \sqrt{\left(x-x+12\right)^2+(2+3)^2} = \sqrt{{12}^2+{5}^2}$$

$$=\sqrt{169}=13$$

I.e, So,

The minimum value is $$\bbox[5px,border:2px solid red]{13}$$

Answer 3

Rewrite the function as $$\sqrt{x^2+4} + \sqrt{(x-12)^2 + 9}$$ This looks like the sum of distances of a point $(x,0)$ from the points (0,2) and $(12,3)$. This can also be thought of as a light ray originating from $(0,2)$, being reflected from the x-axis and striking the point $(12,9)$. This line of thought allows us to use Fermat's Principle to find the smallest value, which occurs by equating the angles of incidence and reflection. This gives us the minimum value of $13$ at $x = \frac{24}{5}$.

Here's a small animation I came up with: