My question reads:
At what points in the plane are the level sets of the following functions orthogonal?
$$ \begin{align*} f(x,y) &= x^2 + y^2 - 2xy \\ g(x,y) &= 2y - 3x \end{align*} $$
Initially, my thought is to set $f(x,y)$ and $g(x,y)$ equal to some constants, $c$ and $d$. From there, I'm unsure of where to go.
What should be my next steps?
Note that a level set $f(x,y)=c$ consists of a pair of lines $x-y = \sqrt{c}$ and $x-y=-\sqrt{c}$, if $c>0$. The level sets $g(x,y)=d$ are lines $2y-3x=d$. These lines intersect nowhere orthogonally (since the product of their slopes is not $-1$).