[Math] Find $f(x,y)$ if each level curve $f(x,y)=C$ is a circle centered at the origin and having radius

Question

Find $f(x,y)$ if each level curve $f(x,y)=C$ is a circle centered at the origin and having radius

(a) $C$

(b) $C^2$

(c) $\sqrt{C}$

(d) $\ln C$.

I am not good at this…

Answer 1

There don't seem to be many options:

$$f(x,y)=x^2+y^2-D\;,\;\;0<D\;\;\text{a constant, and then for example:}$$

$$(a)\;\;\;f(x,y)=C\iff x^2+y^2=C+D\iff D=0$$

$$(b)\;\;\;f(x,y)=C\iff x^2+y^2=C+D=C^2\iff D=C(C-1)$$

and etc. Now you try the other ones.