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$x^4-3x^2+2x^2y^2-3y^2+y^4+2=0$ represents

  1. A pair of circles having the same radius
  2. A circle and an ellipse
  3. A pair of circles having different radii
  4. none of the above
in Others by Veteran (425k points)
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1 Answer

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$\begin{align}x^4-3x^2+2x^2y^2-3y^2+y^4+2&=0\\ \Rightarrow x^4+2x^2y^2+y^4-3(x^2+y^2)+2&=0\\ \Rightarrow (x^2+y^2)^2-3(x^2+y^2)+2&=0\\ \Rightarrow A^2-3A+2&=0 ~; ~[\mathrm{Let~}A=x^2+y^2] \\ \Rightarrow (A-1)(A-2)&=0 \\ \Rightarrow A&=1,2 \end{align}$

$\therefore x^2+y^2=1$ or $x^2+y^2=2$. So there are two circles with different radii.

 

So the correct answer is C.

by Active (3.2k points)

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