plotting of level curves using pen and paper

Question

I want to plot the level curve for the function $f(x,y)=\frac{y^2-x^4}{y^2+x^4}$ . I tried by substituting $f(x,y)=k$. But I am Unable to draw it using paper and pen. Kindly help me.

Answer 1

If $$\frac{y^2-x^4}{y^2+x^4}=k$$ solve for $y^2$ first; this gives $$y^2=x^4\space\frac{1+k }{1-k}$$ So, $$y=\pm x^2\sqrt{\frac{1+k }{1-k}}$$ So, two parabolas which exist only if $\cdots$

Answer 2

$$ \frac{y^2-x^4}{y^2+x^4}=k\implies y^2=\frac{1+k}{1-k}\,x^4,\quad k\ne1. $$ Do you recognize those curves?