# Conics

|      |       |       |

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 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 = at2y = 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.

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

### 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

$\frac{b^2}{a}$

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 ($\pm \ ae, 0$)
• the directrix are at x = $\pm \frac{a}{e}$
• the transverse axis of of length 2a
• the conjugate axis is of length 2b
• the semi latus rectum is of length b2/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 hyperbola so as to makes its asymptotes into the coordinates axes, changes the formula to

xy = c2

where c2 = (a2/2)

Moire Patterns