= Solution
A <projective quadric> is the zero locus of a nonzero homogeneous <quadratic polynomial> $q(v)=v^TBv$, with $B$ symmetric over a <field> of <characteristic> different from two. A <projective point> on it is singular when all first <partial derivatives> vanish. Since $\nabla q(v)=2Bv$, the quadric is nonsingular exactly when $B$ is invertible.
Over the <complex numbers>, a nonzero <quadratic form> in three variables can be diagonalized by an invertible <change of basis>. Its rank classifies a <projective conic>:
* Rank three gives a nonsingular conic.
* Rank two gives \b[two distinct <projective lines>], since the reduced form $X^2+Y^2$ factors as $(X+iY)(X-iY)$. Their intersection is the singular point.
* Rank one gives \b[a double <projective line>], defined by a squared linear form. The gradient vanishes along that line.
The zero polynomial is excluded from the definition of a conic. Thus these last two cases are precisely the singular conics.
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