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
of different types depending whether:
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
|
| [画像:conic section hyperbola] | [画像:conic section hyperbola north south] |
Parametric equations x = a cosh t x = a/cos t |
For information about trig functions: cosh,tanh,cos,tan see this page.
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
|
| [画像:circle] | [画像: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:
- aperture =2θ
