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 


parabola  y² = 4 a x  
hyperbola 

These types can all be visualised as conic sections.
Equations of Hyperbola
eastwest  northsouth  




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 


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