Equation of a conic.

Question

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 ) ?

Answer 1

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 $$ x^2 +2xy-14y^2-5x+40y-24=0\ . $$