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
|x=||-b ± √(b² - 4ac )|
of different types depending whether:
|b² - 4ac > 0||
two real solutions
|b² - 4ac = 0||
|b² - 4ac < 0||
complex number solutions
So our quadratic equation in two variables has different types of solution.
|circle||x² + y² = r²|
|parabola||y² = 4 a x|
These types can all be visualised as conic sections.
Equations of Hyperbola
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.
|x² + y² = r²||
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.
For comparison with above the parametric equations are:
x = a cosθ
y = b sinθ
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)
- aperture =2θ