# Conics

## Contents

### Parabola $e = 1$

A parabola with its center at $(\alpha, \beta)$, focus at $(a + \alpha, \beta)$, directrix at $x = -a + \alpha$ and branches along $+x$ :
$$\begin{array}{l} \color{red}{(y - \beta)^2 = 4a(x - \alpha)} \\ \left.\begin{array}{l} x = at^2 + \alpha \\ y = 2at + \beta \end{array} \right\} \quad \frac{dy}{dx} = \frac{y'}{x'} = \frac{1}{t} \\ \end{array}$$

### Ellipse $e < 1$

An ellipse with its center at $(\alpha, \beta)$, focuses at $(\pm ae + \alpha, \beta)$, directrixes at $x = \pm \frac{a}{e} + \alpha$ , and radii $a$ and $b$ :
$$\begin{array}{l} \color{red}{\frac{(x - \alpha)^2}{a^2} + \frac{(y - \beta)^2}{b^2} = 1} \qquad b = a \sqrt{1 - e^2} \qquad e = \frac{\sqrt{a^2 - b^2}}{a} \\ \left.\begin{array}{l} x = a \cos t + \alpha \\ y = b \sin t + \beta \end{array} \right\} \quad \frac{dy}{dx} = \frac{y'}{x'} = -\frac{b \cos t}{a \sin t} \qquad r = \frac{a(1 - e^2)}{1 + e \cos \theta} \\ \end{array}$$

### Hyperbola $e > 1$

A hyperbola with its center at $(\alpha, \beta)$, focuses at $(\pm ae + \alpha, \beta)$, directrixes at $x = \pm \frac{a}{e} + \alpha$ , distance between two arms $2a$, and asymptotes $y = \pm \frac{b}{a} (x - \alpha) + \beta$ :
$$\begin{array}{l} \color{red}{\frac{(x - \alpha)^2}{a^2} - \frac{(y - \beta)^2}{b^2} = 1} \qquad b = a \sqrt{e^2 - 1} \qquad e = \frac{\sqrt{a^2 + b^2}}{a} \\ \left.\begin{array}{l} x = a \sec t + \alpha \\ y = b \tan t + \beta \end{array} \right\} \quad \frac{dy}{dx} = \frac{y'}{x'} = \frac{b}{a \sin t} \\ \end{array}$$

### Rotation of axes

Any parabola, ellipse or hyperbola can be described by $Ax^2 + By^2 + Cx +Dy + Exy + F = 0$ , where $A, B \neq 0$ , and can be reduced to standard form easily. A rotation of axes is required, if $E \neq 0$ , though.
Let's assume, that we have a point at $(x, \, y)$, which in polar coordinate system is at $(r \cos \theta, \, r \sin \theta)$. Now, the point stays where it is, but the coordinate system is rotated clockwise by $\alpha$ . After that, point's new coordinates are:
\begin{align} \color{red}{(X, \, Y)} &= (r \cos (\theta + \alpha), \, r \sin (\theta + \alpha)) \\ &= (r \, (\cos \theta \cos \alpha - \sin \theta \sin \alpha), \, r \, (\cos \theta \sin \alpha + \sin \theta \cos \alpha)) \\ &= \color{red}{(x \cos \alpha - y \sin \alpha, \, x \sin \alpha + y \cos \alpha)} \end{align}
How is this useful? It gives an opportunity to neutralize that $xy$ , which prevents the reduction to the standard form, by substituting all $x$-s and $y$-s with the values shown above and remembering, that everything is rotated clockwise by $\alpha$ .