## Conic Sections

The curves produced by cutting a double cone with a plane are called conic sections. These curves are the boundaries of the sections exposed by the cut.

- When the plane making the cut is perpendicular o the axis, the curve produced is a circle.
- When the cutting plane is inclined at an angle between the horizontal and that of the generator, the boundary curve is an ellipse.
- When the cutting plane is parallel to a generator the resulting curve is a parabola.
- If the cutting plane meets both parts of the cone at an angle the curve produced is in two branches and is a hyperbola.

The conics of various kinds have a common property : There is a fixed point called the focus and a fixed line called the directrix, such that for any point on the conic, the distance to the focus and the distance to the directrix have a constant ratio called the eccentricity, i.e. *FP/PN = e
*

The circle can be regarded as the limiting case of an ellipse as the directrix moves further from the origin, e approaches 0, and the focii approach the centre of the ellipse.

Standard Forms of Conics |
||||

Conic |
Circle |
Parabola |
Ellipse |
Hyperbola |

Eccentricity |
e = 0 |
e = 1 |
0 < e < 1 |
e > 1 |

Standard Equation |
x^{2} + y^{2} = a^{2} |
y^{2} = 4ax |
(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1 |
(x^{2}/a^{2}) – (y^{2}/b^{2}) = 1 |

Tangent to |
x_{1} + y_{1y} = a^{2} |
y_{1}y = 2a(x + x_{1}) |
(x_{1}x/a^{2}) + (y_{1}y/b^{2}) = 1 |
(x1x/a2) – (y1y/b2) = 1 |

The parametric equations of a circle are, *x* = *a* cos *θ* and *y* = *a* sin *θ*. As *θ* takes the values in the range 0 ≤ *θ* < 360° then a point moves around the circle in an anticlockwise direction. A variable such as *θ* which is used to determine the position of a point on a curve is known as a parameter. In many applications to physics the natural parameter to choose would be time, so that the position of a point is determined by the value of the time.

The parametric equations for a parabola are, x = *at*^{2} and *y* = 2*at*. The equation of the normal to the parabola is, *y* + *tx* = 2*at* + *at*^{3}. The polar form of a parabola is *L/r* = 1 + cos *θ* (for -π < *θ* < π)

The parametric equations for an ellipse are, *x* = *a* cos *θ* and *y* = *b* sin *θ*, and an alternate (and often more useful) form of the parametric equations are, (cos *θ*_{1}/*a*)*x* + (sin *θ*_{1}/*b*)*y* = 1

There is one feature of the hyperbola that is not present in any of the other conics. The branches of the hyperbola are ‘hemmed in’ by the lines, *y* = ± (*b*/*a*)*x*. The hyperbola approaches one of these lines ever more closely but does not touch or cut it. These lines are called the **asymptotes**. The parametric equations for a hyperbola are, cosh *x* = (*e*^{x} + *e*^{-x})/2 and sinh *x* = (*e*^{x} + *e*^{-x})/2

The general equation,

*Ax*^{2} + 2*Hxy* + *By*^{2} + 2*Gx* + 2*Fy* + *C* = 0

Includes all the different types of conics. When,

*H*= 0,*A*=*B*≠ 0 → Circle*H*^{2}=*AB*→ Parabola*H*^{2}<*AB*→ Ellipse*H*^{2}>*AB*→ Hyperbola