A template solution for the minimum of an integral

Question

I came across a question involving the minimization of the integral: $$\int_{\theta_{1}}^{\theta_{2}}{\sqrt{1+(\frac{dz}{zd\theta})^2}}d\theta$$ (Where $\theta_{1}$ and $\theta_{2}$ are fixed numbers and the function $z (\theta)$ can vary with fixed boundary conditions, ie. $z(\theta_1) = z_1$ and $z(\theta_2) = z_2$, where again, $z_1$ and $z_2$ are fixed numbers). The key idea was to make a "coordinate transformation" $z\to r$ such that $\frac{dz}{zd\theta} = \frac{dr}{d\theta}$. This gives us $r = lnz +c$ and the solution for the resultant integral is a straight line in the new coordinates. I was wondering if we can generalise the result so that the minimum value of: $$\int_{a}^{b}{\sqrt{1+(f(x,y)\frac{dy}{dx})^2}dx}$$ comes with $y$ varying such that: $f(x,y)\frac{dy}{dx} = c_1$ where $c_1$ and $c_2$, the constant associated with integrating the differential equation can be obtained by applying a pair of given boundary conditions. On ascertaining the viability of such a relation between x and y, the minimum value is: $$\int_{a}^{b}{\sqrt{1+c_1^2}dx} = \sqrt{1+c_1^2}\int_{a}^{b}{dx} = \sqrt{1+c_1^2}(b-a)$$