Question

What are represented by the equation $(x^2-a^2{)}^2-y^4=0.$

Answer 1

${(x^2-a^2)}^2-y^4{(x^2-a^2)}^2-{\left(y^2\right)}^2=0$

Factorising the equation

$(x^2-a^2+y^2)(x^2-a^2-y^2)=0$

Then, either $x^2-a^2+y^2=0$ or $x^2-a^2-y^2=0$

If $x^2-a^2+y^2=0$

$x^2+y^2=a^2$

represents a circle centred at origin and radius $a$

If $x^2-a^2-y^2=0$

$x^2-y^2=a^2$

represents a rectangular hyperbola.

Thus the combined equation represents a circle and a rectangular hyperbola.