The is a problem from Differential Equations with Applications and Historical Notes by George F. Simmons. The chapter is about the calculus of variation.
The hint is to represent equation of a right circular cone $z^2 = a^2(x^2+y^2)$ where $z \geq0$ parametrically by using
$$x=\frac{rcos(\theta\sqrt{1+a^2})}{\sqrt{1+a^2}}$$ $$y=\frac{rsin(\theta\sqrt{1+a^2})}{\sqrt{1+a^2}}$$ $$z=\frac{ar}{\sqrt{1+a^2}}$$
Show that parameters $r$ and $\theta$ represent ordinary polar coordinates on the flattened cone. And then show that a geodesic $r = r(\theta)$ is a straight line in these polar coordinates.
I don't know that to get the equation for right circular cone into polar coordinates. Just plugging those for $x,y$ and $z$ into equation of a right circular cone won't do the trick. I just get
$$\frac{a^2r^2}{1+a^2} = \frac{a^2r^2}{1+a^2}$$
Also showing that we get polar coordinates on the flattened cone is a problem.
I guess that to show that geodesic is a straight line, I have to solve Euler-Lagrange equation. But this is the last step of a problem and I don't know how to get there.
