Conics

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Conics (Conic Sections, in full) are defined as the intersection of a plane and a cone (and, at a simple level, we just consider the four shapes obtained when the plane does not pass thru the vertex of the cone).

Alternatively : - If S is a fixed point and l is a fixed line which does not pass thru S, a conic is the locus of points such that the ratio of the distance from S to the distance from l is a constant. The fixed point is the focus, and the fixed line is the directrix. The constant ratio is called the eccentricity.


Summary of Basic Properties

Circle

Ellipse

Parabola

Hyperbola

Standard Cartesian Equation :

x2 + y2 = r2

Formula for Ellipse

y2 = 4ax

Formula for Hyperbola

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

Parametric Representation for Ellipse

x = at2
y = 2at

Parametric Formulas for Hyperbola

or

Parametric Formulas for Hyperbola

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

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

General Conic Formula in Polar Form

where L is the semi latus rectum.

The latus rectum is a chord passing through the focus, parallel to the directrix.


Ellipse

The cartesian equation of an ellipse is

Formula for Ellipse
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

From the general polar form, the equation for an ellipse is

Formula for Ellipse in Polar Form

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.

Ellipse - Signal from Focus to Focus


Hyperbola

The cartesian equation of an hyperbola is

Formula for Hyperbola

In its general form, with the origin at the center of coordinates

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

Formula for Hyperbola, rejigged

As x becomes larger, y tends to

Asymptotes of Hyperbola

these are the equations of the asymptotes.

Rectangular Hyperbola
A hyperbola is rectangular if its asymptotes are perpendicular.

From

Asymptotes of Hyperbola

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 hyperbola so as to makes its asymptotes into the coordinates axes, changes the formula to

xy = c2

where c2 = (a2/2)


Moire Patterns