Question

What loci are represented by the equations :
$(x^2-1{)}^2+(y^2-4{)}^2=0.$

Answer 1

${(x^2-1)}^2+{(y^2-4)}^2=0{(y^2-4)}^2=-{(x^2-1)}^2$

Both the side of equation are perfect saqure. A perfect saqure can only be equal to negative of other perfect saqure if both L.HS and R.H.S of the equation are zero.

L.H.S is zero at $y=2,-2$ and R.H.S is zero at $x=1,-1$

So the given equation only represents four points i.e. $(1,2),(1,-2),(-1,2)$ and $(-1,-2)$