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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. –  Sasha Aug 30 '11 at 20:32
    
Look at the classification by discriminant section –  marwalix Aug 30 '11 at 20:36
    
@Sasha,marwalix Thanks to you both... –  Kuku Aug 30 '11 at 20:47
1  
Don't you mean $p = p_1 + ip_2$ instead of $p = p_1 + ip_1$, and so on? –  Rahul Aug 30 '11 at 21:44
    
@Rahul Yes! you are right...Thanks. –  Kuku Aug 30 '11 at 21:51
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