Question

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

Answer 1

${(x^2-a^2)}^2{(x^2+b^2)}^2+c^4{(y^2-a^2)}^2=0$
This being the sum of two perfect squares, each term must be zero
${(x^2-a^2)}^2{(x^2-b^2)}^2=0$ or $(x^2-a^2)(x^2-b^2)=0$
or $(x-a)(x+a)(x-b)(x+b)=0$
and $c^4{(y^2-a^2)}^2=0$
or $c^2{(y^2-a^2)}^2=0$
$\because c\ne 0\therefore y=\pm a$
$\therefore x=\pm a,\pm b$
$y=\pm a$
As both of these must simultaneously satisfy, the given line represents $8$ points which we get as a result of different combination of $\left(1\right)$ and $\left(2\right)$ , namely
$(\pm a,\pm a),(\pm b,\pm a)$