# Cones and Conic Sections

[latexpage]

## Relevant terminology

Generator = a linear line (green line) that intersects a vertical line called the axis at a fixed point (purple dot) called the vertex

• Note: the angle between the generator and the axis is called the vertex angle
• Rotating the generator 360° around the vertex and tracing out the rotation into a 3D surface gives a cone

Double right circular cone = a cone whose bases (the top and bottom faces of the cone) are perpendicular to the vertical axis and form a perfect circle

•  Directrix = 2D outline of the base
• The directrix of a right circular cone is always a perfect circle
• Nappe = lateral surface of a single circular cone
• For the diagram above,
• The upper nappe is outlined in pink
• The lower nappe is outlined in blue

## Overview of Conic Sections

conic section is the 2D curve corresponing to a particular 2D “slice” of a 3D cone

• The angle at which the 2D plane slices the 3D cone determines the type of conic section we get

## Parabolas

General equation:

$y - k = a(x - h)^2$

$x - h = a(y - k)^2$

## Circles

General equation:

$(x-h)^2 + (y-k)^2 = R^2$

• The center of the circle is defined as $C:(h,k)$
• The radius of the cirlce is defined as $R$

## Ellipses

General equation:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

## Hyperbolas

General equation:

$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

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