Explicit Solution to Euler Lagrange Equation of Shortest Distance between two Points

Question

So if $f(t)=(x(t),y(t))$ with $f(0)=a$ and $f(1)=b$, I should minimize $L(f)=\int_0^1{|\dot{f(t)}|}dt$. I get a jumble of equations when solving the Euler Lagrange equations with respect to $t$. Would someone help explicitly solving them. Please avoid using $y'(x)$ since I am trying to generalize this to n space

Thanks in advanced for any help.

Answer 1

Hint: Consider the squared Lagrangian cf. e.g. this Math.SE post. The shortest curve between $a$ and $b$ is unsurprisingly a straight line.