linguistic:paraphrasing
Three-point form of ellipse equation

For example, for P 1 = (2, 0), P 2 = (0, 1), P 3 = (0, 0) {\displaystyle P_{1}=(2,\,0),\;P_{2}=(0,\,1),\;P_{3}=(0,\,0)} and q = 4 {\displaystyle q=4} one obtains the three-point form

Analogously to the circle case, the equation can be written more clearly using vectors: (x → − x → 1) ∗ (x → − x → 2) det (x → − x → 1, x → − x → 2) = (x → 3 − x → 1) ∗ (x → 3 − x → 2) det (x → 3 − x → 1, x → 3 − x → 2), {\displaystyle {\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)}},}

where ∗ {\displaystyle *} is the modified dot product u → ∗ v → = u x v x + q u y v y. {\displaystyle {\vec {u}}*{\vec {v}}=u_{x}v_{x}+{\color {blue}q}\,u_{y}v_{y}.}