# 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 : | x |
| y |
| ||

Eccentricity (e): | 0 | 0 < e <1 | 1 | 1 < e | ||

Relation between a,b and e | b = a | b | b | |||

Parametric Representation |
| x = at |
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###
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0

The actual type of conic can be found from the sign of B^{2} - 4AC

If B | 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

###
Ax^{2} + 2Bxy + Cy^{2} + Dx + Ey + F = 0

Here the type of conic must be found from the sign of B^{2} - AC

If B | 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

PF_{1} + PF_{2}

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 b
^{2}/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

PF_{1} - PF_{2}

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

x^{2} - y^{2} = 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 = c^{2}

*where c ^{2} = (a^{2}/2)*

Moire Patterns