<s>[INST] Consider two circles S1 and S2 given by $x^4 + y^4 +2x^2y^2-10x^2+6y^2+9=0 $? How can I separate the circles? I recently gave a test and had this question there . To be honest I was confused on seeing this question. I was unable to even deduce the approach.
I am a student studying in class 11 so I don't have access to desmos during tests.
 [/INST] Rewrite the equation as
$$(x^2+y^2)^2-16x^2+6x^2+6y^2+9=0$$
which, using the difference of two squares, is equivalent to
$$(x^2-4x+y^2)(x^2+4x+y^2)+6(x^2+y^2)+9=0.$$
Then rewrite $6(x^2+y^2)$ as $3(x^2-4x+y^2)+3(x^2+4x+y^2)$ to enable us to factorise.
Now let $a=x^2-4x+y^2,b=x^2+4x+y^2$. Then we have $ab+3a+3b+9=0$, which factorises as $(a+3)(b+3)=0$. Hence the equations of the circles are $a+3=0$ and $b+3=0$, or
$$(x-2)^2+y^2=1, (x+2)^2+y^2=1.$$
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