Sketch the level curves of the function

Question

$f(x,y)= (y-3x)^2$

let $k=0$

then $z=(y-3x)^2 \to 0=y^2-6xy+9x^2$

This is as far as I can get is my approach incorrect? Is this a special case of a polynomial degree two?

maybe $0=y^2-6xy+9x^2-z$

Answer 1

The way I would do it would be to break up the level curves into a "top" and "bottom" half:

$$(y-3x)^2 = k \text{ (constant) }\implies \begin{cases} y - 3x = +\sqrt{k} \\ y- 3x = -\sqrt{k} \end{cases}$$

It will be nicer to compute if you pick your $k$'s to be perfect squares.