# Maths - Hyperbola and Parabola

### Quadratic Equations in Two Variables

We can represent a general quadratic equation in two variables as:

A x² + B xy + C y²+ D x + E y + F = 0

In the same way that the quadratic equation in one variable:

a x² + b x + c = 0

has solutions

 x= -b ± √(b² - 4ac ) 2a

of different types depending whether:

 b² - 4ac > 0 two real solutions b² - 4ac = 0 one solution b² - 4ac < 0 complex number solutions

So our quadratic equation in two variables has different types of solution.

circle x² + y² = r²
ellipse
 x² + y² =±1 a² b²
parabola y² = 4 a x
hyperbola
 x² - y² =±1 a² b²

These types can all be visualised as conic sections.

### Equations of Hyperbola

east-west north-south
 x² - y² =1 a² b²
 x² - y² = -1 a² b²

Parametric equations

x = a cosh t
y = b tanh t

x = a/cos t
y = b/tan t

Hyperbola Focal Points

## Equations of Parabola

y² = 4 a x

This can be represented by the intersection of the cone and a plane which is parallel to the face of the cone.

## Equations of Circle and Ellipse

An ellipse is a circle that may be expanded differently in the x and y directions. Or, to reverse the argument, a circle is an ellipse whose extent is equal in both dimensions.

Circle Ellipse
x² + y² = r²
 x² + y² =±1 a² b²

When we intersect the cone with a plane parallel to its base we get a circle, when we intersect at an angle (But less than the angle of the cone face) then we get an ellipse.

### Parametric equations

For comparison with above the parametric equations are:

x = a cosθ
y = b sinθ

## Conic Sections

The equation for a cone in 3 dimensions is:

(x² + y²)cos²θ - z² sin²θ

Or in terms of parametric equations:

x = u cos(θ) cos(t)
y = u cos(θ) sin(t)
z = u sin(θ)

where:

• aperture =2θ