Level Curves Question

Question

How do I go about sketching the level curve $m(x,y) = \frac{y}{x^2 + 2y^2}$? I'm struggling to determine how to go about it. Thank you.

Answer 1

Adjust constant into the form $$ \frac{y}{x^2+2y^2}=\frac{1}{4c} $$ Complete the square $$ x^2+2 y^2-4cy=0,\quad x^2+2(y^2-2cy+c^2)=2 c^2 $$ $$ \frac{x^2}{(\sqrt{2 c})^2}+\frac{(y-c)^2}{c^2} =1 $$ an ellipse in its standard form.