Smooth conics in complex projective plane

Question

How can I show that each smooth conic in $\Bbb P _2\Bbb C $ is isomorph to the conic given by the equation $x^2+y^2+z^2=0$? I thought about considering the quadratic form $A$ associated to my conic, A is symmetric so I can say that $D=P^{-1}AP$ where D is diagonal with coefficients $d_1, d_2, d_3$. Know I probably have to define a maps from $\Bbb P _2\Bbb C $ to $\Bbb P _2\Bbb C $ using $P$, but I can't understand which is the correct way to do that! Can anyone help me? Thanks in advance!