Length of the shortest road that goes from A to the highway and then on to B The following problem is from George Simmons' Calculus with Analytic Geometry, second edition, chapter $4$, section $4$, problem $15$:
Two towns, $A$ and $B$, lie on the same side of a straight highway. Their distance apart is $c,$ and their distances from the highway are $a$ and $b.$ Show that the length of the shortest road that goes from $A$ to the highway and then on to $B$ is $\sqrt{c^{2} + 4ab}.$
Is the length $\sqrt{c^{2} + 4ab}$ correct? I believe this is an erratum, as I obtain $\sqrt{c^{2} + (a+b)^{2}},$ when I use $x = \frac{ac}{a+b}$ in the length formula $L = \sqrt{a^{2} + x^{2}} + \sqrt{b^{2} + (c-x)^{2}}.$
The specified value of $x$ was found via Calculus and Heron's reflection technique (using similar triangles). WolframAlpha computes the same length.

You have a typo. Instead of $c$ (which is the distance between $A$ and $B$) in your formula you should have $d$ which is the difference in the $x$ coordinates of points $B$ and $A$. It solves
$$\sqrt{d^2+(b-a)^2}=c$$
$$d^2=c^2-(b-a)^2$$
Now you plugging this $d$ on place of your $c$ we get
$$\sqrt{c^2-(b-a)^2+(a+b)^2}=\sqrt{c^2+4ab}$$
Which is the right answer.
