Ellipse
Tangent

An arbitrary line g {\displaystyle g} intersects an ellipse at 0, 1, or 2 points, respectively called an exterior line, tangent and secant. Through any point of an ellipse there is a unique tangent. The tangent at a point (x 1, y 1) {\displaystyle (x_{1},\,y_{1})} of the ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} has the coordinate equation: x 1 a 2 x + y 1 b 2 y = 1. {\displaystyle {\frac {x_{1}}{a^{2}}}x+{\frac {y_{1}}{b^{2}}}y=1.}

A vector parametric equation of the tangent is: x → = (x 1 y 1) + s (− y 1 a 2 x 1 b 2), s ∈ R. {\displaystyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s\left({\begin{array}{r}-y_{1}a^{2}\\x_{1}b^{2}\end{array}}\right),\quad s\in \mathbb {R}.}

Proof: Let (x 1, y 1) {\displaystyle (x_{1},\,y_{1})} be a point on an ellipse and x → = (x 1 y 1) + s (u v) {\textstyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s{\begin{pmatrix}u\\v\end{pmatrix}}} be the equation of any line g {\displaystyle g} containing (x 1, y 1) {\displaystyle (x_{1},\,y_{1})}. Inserting the line's equation into the ellipse equation and respecting x 1 2 a 2 + y 1 2 b 2 = 1 {\textstyle {\frac {x_{1}^{2}}{a^{2}}}+{\frac {y_{1}^{2}}{b^{2}}}=1} yields: (x 1 + s u) 2 a 2 + (y 1 + s v) 2 b 2 = 1 ⟹ 2 s (x 1 u a 2 + y 1 v b 2) + s 2 (u 2 a 2 + v 2 b 2) = 0. {\displaystyle {\frac {\left(x_{1}+su\right)^{2}}{a^{2}}}+{\frac {\left(y_{1}+sv\right)^{2}}{b^{2}}}=1\ \quad \Longrightarrow \quad 2s\left({\frac {x_{1}u}{a^{2}}}+{\frac {y_{1}v}{b^{2}}}\right)+s^{2}\left({\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)=0\.} There are then cases:

Using (1) one finds that (− y 1 a 2 x 1 b 2) {\displaystyle {\begin{pmatrix}-y_{1}a^{2}&x_{1}b^{2}\end{pmatrix}}} is a tangent vector at point (x 1, y 1) {\displaystyle (x_{1},\,y_{1})}, which proves the vector equation.

If (x 1, y 1) {\displaystyle (x_{1},y_{1})} and (u, v) {\displaystyle (u,v)} are two points of the ellipse such that x 1 u a 2 + y 1 v b 2 = 0 {\textstyle {\frac {x_{1}u}{a^{2}}}+{\tfrac {y_{1}v}{b^{2}}}=0}, then the points lie on two conjugate diameters (see below). (If a = b {\displaystyle a=b}, the ellipse is a circle and "conjugate" means "orthogonal".)
keywords:existence, oulipo:avoid_letter[o]