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
+ =±1
parabola y² = 4 a x
hyperbola
- =±1

These types can all be visualised as conic sections.

Equations of Hyperbola

east-west north-south
- =1
- = -1
conic section hyperbola conic section hyperbola north south

Parametric equations

x = a cosh t
y = b tanh t

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

 

For information about trig functions: cosh,tanh,cos,tan see this page.

hyperbola

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.

conic section parabola

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
+ =±1
circle ellipse
   
   

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:


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see also:

 

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