# The foci and rotation angle of an ellipse, $E_0$, as a function of the coefficients of an equation of $E_0$

## TL;DR

This paper derives formulas for the foci and rotation angle of an ellipse based on its algebraic equation coefficients, providing explicit relationships useful for geometric analysis.

## Contribution

It introduces new explicit formulas linking ellipse foci and rotation angle directly to the coefficients of its defining equation.

## Key findings

- Formulas for ellipse foci as functions of equation coefficients
- Precise formula for ellipse rotation angle from coefficients
- Theoretical foundation for geometric properties of ellipses

## Abstract

First, we give a formula for the foci of an ellipse, $E_0$, as a function of the coefficients of an equation of $E_0$(see Theorem <ref>T2</ref>). To prove Theorem <ref>T2</ref>, we use two interesting formulas proven in <cite>B</cite> and in <cite>S</cite>. Our second result(see Theorem <ref>T3</ref>), is a more precise formula for the rotation angle of $E_0$ as a function of the coefficients of an equation of $E_0$.