# Finding level curves to function

I am supposed to find and draw a few level curves for the function $$g(x,y) = e^{\sqrt{x^2-y^2}}$$. I have already calculated the domain of the function: $$Df=\lbrace(x,y) : y ≤ ±|x|\rbrace$$

In order to find a few level curves, I began by calculating the following for a constant c: $$e^{\sqrt{x^2-y^2}}=c$$, This gives $$\sqrt{x^2-y^2}=\ln(c)$$ and $$c>0$$.

The first level curve, when $$c=1$$:

$$g(x,y)=e^{\sqrt{x^2-y^2}}=1\implies x^2-y^2=0\implies y=±x,$$ which I can I can easily draw, but I am having trouble finding more level curves. It feels like they get very complicated e.g. when $$c=1$$.

How do I find values of $$c$$ that result in level curves that aren't too hard to draw?

• Not complicated. Just note that $g(x,y)=c>0 \iff x^2-y^2=(\ln c)^2$, which is just another positive constant. You need to be familiar with curves $x^2-y^2=k>0$. Jan 22 at 1:34

For $$x^2- y^2\ge 0$$ we have
$$g(x,y) = c_1 = e^{\sqrt{x^2-y^2}}\Rightarrow x^2-y^2=c_2=(x+y)(x-y)=c_2$$
so for $$c_2=0$$ we have two lines $$\{x+y=0\}\cup \{x-y=0\}$$ and for $$c_2\ne 0$$ we have a slanted hyperbole.