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...

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 \( ... \) or \[ ... \] 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.