Editing Ellipse (section)

====Three-point form of ellipse equation====

: A consequence, one obtains an equation for the ellipse determined by three non-collinear points <math>P_i = \left(x_i,\, y_i\right)</math>: <math display="block">
\frac{({\color{red}x} - x_1)({\color{red}x} - x_2) + {\color{blue}q}\;({\color{red}y} - y_1)({\color{red}y} - y_2)}
     {({\color{red}y} - y_1)({\color{red}x} - x_2) - ({\color{red}y} - y_2)({\color{red}x} - x_1)} =
\frac{(x_3 - x_1)(x_3 - x_2) + {\color{blue}q}\;(y_3 - y_1)(y_3 - y_2)}
     {(y_3 - y_1)(x_3 - x_2) - (y_3 - y_2)(x_3 - x_1)}\ .
</math>

For example, for <math>P_1 = (2,\, 0),\; P_2 = (0,\,1),\; P_3 = (0,\, 0)</math> and <math>q = 4</math> one obtains the three-point form
: <math>\frac{(x - 2)x + 4y(y - 1)}{yx - (y - 1)(x - 2)} = 0</math> and after conversion <math>\frac{(x - 1)^2}{2} + \frac{\left(y - \frac{1}{2}\right)^2}{\frac{1}{2}} = 1.</math>

Analogously to the circle case, the equation can be written more clearly using vectors:
<math display="block">
    \frac{\left({\color{red}\vec x} - \vec x_1\right)*\left({\color{red}\vec x} - \vec x_2\right)}
         {\det\left({\color{red}\vec x} - \vec x_1,{\color{red}\vec x} - \vec x_2\right)}
  = \frac{\left(\vec x_3 - \vec x_1\right)*\left(\vec x_3 - \vec x_2\right)}
         {\det\left(\vec x_3 - \vec x_1, \vec x_3 - \vec x_2\right)},
</math>

where <math>*</math> is the modified [[dot product]] <math>\vec u*\vec v = u_x v_x + {\color{blue}q}\,u_y v_y.</math>