Summary of Basic Properties
| Circle | Ellipse | Parabola | Hyperbola |
| Standard Cartesian Equation : | x2 + y2 = r2 |  | y2 = 4ax |  |
| Eccentricity (e): | 0 | 0 < e <1 | 1 | 1 < e |
| Relation between a,b and e | b = a | b2 = a2(1-e2) |
| b2 = a2(e2-1) |
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| Parametric Representation |
|  | x = at2
y = 2at |
or
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| Definition : It is the locus of all points which meet the condition... | distance to the origin is constant | sum of distances to each focus is constant | distance to focus = distance to directrix | difference between distances to each foci is constant |
It might tidy the logic up to consider a circle to be a special case of an ellipse. Then there are two 'main' classes
- an ellipse, with e < 1
- a hyperbola, with e > 1
and a 'critical' class - the parabola with e = 1.
The General Equation of a Conic
The General Equation for a Conic is
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
The actual type of conic can be found from the sign of B2 - 4AC
| If B2 - 4AC is... | then the curve is a... |
| < 0 | ellipse, circle, point or no curve. |
| = 0 | parabola, 2 parallel lines, 1 line or no curve. |
| > 0 | hyperbola or 2 intersecting lines. |
note : the above notation brings a close analogy with the formulas of quadratic equations. Sometimes, however, the formula is stated slightly differently
Ax2 + 2Bxy + Cy2 + Dx + Ey + F = 0
Here the type of conic must be found from the sign of B2 - AC
| If B2 - AC is... | then the curve is a... |
| < 0 | ellipse, circle, point or no curve. |
| = 0 | parabola, 2 parallel lines, 1 line or no curve. |
| > 0 | hyperbola or 2 intersecting lines. |
Polar Form
For an origin at a focus, the general polar form (apart from a circle) is
where L is the semi latus rectum.
Ellipse
The cartesian equation of an ellipse is
where a and b would give the lengths of the semi-major and semi-minor axes.
In its general form, with the origin at the center of coordinates
- the foci are at
- the directrix are at
- the major axis of of length 2a
- the minor axis is of length 2b
- the semi latus rectum is of length
From the general polar form, the equation for an ellipse is
For any point P on the perimeter, the sum
PF1 + PF2
will be constant, no matter which point is chosen as P.
Hence, an ellipse can also be defined as the locus of a point which moves in a plane so that the sum of its distances from two fixed points is constant.
According to Kepler's First law, the orbit of a planet is an ellipse.
The Earth is shaped like an ellipsoid.
| Any signal from one of the foci will pass thru the other focus. |  |
Hyperbola
The cartesian equation of an hyperbola is
In its general form, with the origin at the center of coordinates
- the foci are at (+/- ae, 0)
- the directrix are at x = +/- a/e
- the transverse axis of of length 2a
- the conjugate axis is of length 2b
- the semi latus rectum is of length 2b2/a
Note the similarity in notation with ellipses; although now the eccentricity is greater than one
Also by analogy with an ellipse
For any point P on a hyperbola, the sum
PF1 - PF2
will be constant, no matter which point is chosen as P.
Hence, a hyperbola can also be defined as the locus of a point which moves in a plane so that the difference of its distances from two fixed points is constant.
Asymptotes of Hyperbola
Rejigging the hyperbola formula to
As x becomes larger, y tends to
these are the equations of the asymptotes.
Rectangular Hyperbola
A hyperbola is rectangular if its asymptotes are perpendicular.
From
this requires
b = a
Substituting this into the cartesian formula for a hyperbola produces
x2 - y2 = 1
which has an eccentricity equal to the square root of 2
Rotating a rectangular so as to makes its asymptotes into the coordinates axes, changes the formula to
xy = c2
where c2 = (a2/2)
