SOURCE : http://bdaugherty.tripod.com/moon/conics.html

 

 

 

 

 

 

  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.

 

 

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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.

 

 

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Polar Form

 

For an origin at a focus, the general polar form (apart from a circle) is

 

 

 

 

where L is the semi latus rectum.

 

 

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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.

 

 

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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)