Existence : The curve is well defined for $1-x^2 >0$ ie $x^2 <1$
$=>x < 1\;and\; x > -1$
(ii) Symmetry: The curve is symmetric about x-axis and y-axis and hence about the orgin
(iii) Asymptotes: It has no asymptotes
Loops: for $-1 < x < 0 \;and \;0 < x < a,y^2 >0$
=> y takes corresponding +ve and -ve values.
The curve passes through (0,0) turns and once throu(-1,0) and (1,0)
$\therefore $ a loop is formed between $x=-1\;and\; x=0$ and another loop is formed between $x=0\;and \;x=1$