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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)
Parametric Representation			x = at² y = 2at	or
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.

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)

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