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If $p,q,r\in \mathbb{C}$, how would one describe the curve $$\mathrm{Re}\left(pz^2+qz+r\right)=0.$$

If I write $p=p_1+ip_2$, $q=q_1+iq_2$, $r=r_1+ir_2$, and $z=x+iy$, then

$$\mathrm{Re}\left(pz^2+qz+r\right)=p_1\left(x^2-y^2\right)-2p_2xy+q_1x-q_2y+r_1=0.$$

Can I therefore say that the above is a polynomial in $x$ and $y$? If it is right, would that be enough? Are there any cases, I must consider?

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    Yes, the right-hand-side of your equation is quadratic polynomial in each $x$ and $y$. The curve thus obtained is called [conic section](http://en.wikipedia.org/wiki/Conic_section).2011-08-30
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    Look at the classification by discriminant section2011-08-30
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    @Sasha,marwalix Thanks to you both...2011-08-30
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    Don't you mean $p = p_1 + ip_2$ instead of $p = p_1 + ip_1$, and so on?2011-08-30
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    @Rahul Yes! you are right...Thanks.2011-08-30
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    @Kuku, one of your p_1 should be p_2 and q_1(x-y) should be q_1x-q_2y.2011-09-01
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    @Didier, Yes!...you are right... I'll fix it.2011-09-03

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