Let's consider the simplest parabola, which can be represented by the space of points $[x,y]$ in $\mathbb{A}^2$ over an arbitrary field $\mathbb{F}$ with $\text{char}(\mathbb{F}) \neq 2$ that satisfies the equation $y = x^2$. Note that a general affine transformation on this parabola will result in a general parabola.

To compute the tangent line of the parabola at the point $P \equiv \left[ t,t^2 \right]$ we pick another point $Q \equiv \left[ u,u^2 \right]$ on the parabola, so that the line joining $P$ and $Q$ has vector equation

$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} t \\ t^2 \end{pmatrix} + \lambda \begin{pmatrix} u-t \\ u^2-t^2 \end{pmatrix},$

for $\lambda \in \mathbb{F}$. Noting that $u^2 - t^2 = (u-t)(u+t)$, we use the fact that $x - t = \lambda (u-t)$ to obtain

$y - t^2 = \lambda \left( \frac{x-t}{\lambda} \right) (u+t),$

which simplifies to

$y = (u+t)x - ut.$

This line is called the **secant line** or the **secant chord**. If $Q=P$, then $u = t$ and the secant chord has equation $y = 2tx - t^2$; this line is then called the **tangent line** of the parabola at the point $P$. We now have a methodology for exactly computing tangent lines to parabolas which avoids limiting processes. Note that this also implies that the **vertex** of the parabola, i.e. the point at which the tangent line of the parabola takes the form $y = c$ for a constant $c \in \mathbb{F}$, is $[0,0]$

As an example, we consider the general affine transformation $T:\mathbb{A}^2 \mapsto \mathbb{A}^2$ given by

$T(\begin{pmatrix} x \\ y \end{pmatrix}) = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}+ \begin{pmatrix} f \\ g \end{pmatrix}.$

The vertex $[0,0]$ is mapped by $T$ to $[f,g]$, which is the vertex of the translated parabola. I will leave it as an exercise to figure out

- the equation of the tangent line; and
- hence, the equation of the general parabola.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

$</code> ... <code>$</code>...<code>."> Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in $</span> ... <span>$ or $</span> ... <span>$ to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

There are no comments in this discussion.