Equation of a conic. I have an question that is confusing me, I tried to search on internet but no clear answer...
Questions : How can i determine the equation of a conic given 4 or 5 points example : Given the points 
A(1,1)
B(0,2)
C(-3,0)
D(2,1)
E(1,2)
How to write the equation of conic passing through the points ( A,B,C,D ) ?
 A: The general equation of a plane conic has the form $\ ax^2 + bxy + cy^2 + dx + ey + f = 0\ $.  Substituting the coordinates of any point lying on the conic into this equation gives you a homogeneous linear equation in the coefficients $\ a, b, c, d, e\ $ and $\ f\ $. Since there are six unknown coefficients you need at least five equations to determine them as unique multiples of a single non-zero parameter (and therefore to determine the conic uniquely).  For the given example, we have:
$$
\begin{matrix}
\mathbf{A} \mbox{ on the conic }&\implies& a + b + c + d + e + f &=& 0\\
\mathbf{B} \ \mbox{ on the conic }&\implies& 4c + 2e + f &=& 0\\
\mathbf{C} \ \mbox{ on the conic }&\implies& 9a -3d +f &=& 0\\
\mathbf{D} \ \mbox{ on the conic }&\implies& 4a+ 2b + c + 2d + e + f &=& 0\\
\mathbf{E} \ \mbox{ on the conic }&\implies& a + 2b + 4c + d + 2e + f &=& 0\ .
\end{matrix}
$$
These equations have solution
$$
\left(a,b,c,d,e,f\right) = -\frac{f}{24}\,\left(1, 2, -14, -5, 40, -24\right)\ ,
$$
So the conic passing through these five points has equation
$$
x^2 +2xy-14y^2-5x+40y-24=0\ .
$$
