Conics Sections
This book will guide you through the basic ideas of conic sections
Table of Contents
- Circles
- Circles
- Circles: Expanded Form
- Expanded Form: Complete the Square
- Circles: Completing the Square, Final Solution
- Ellipses
- Ellipse Introduction
- Vertices of an Ellipse
- Foci Equation
- Foci of Ellipses
- Hyperbola
- Hyperbola Introduction
- Horizontal Hyperbola Example
- Vertical Hyperbola
Circles
Introduction
The standard equation of a circle is given by: [math]\left(x-h\right)^2+\left(y-k\right)^2=r^2[/math] [br]This should remind you of the pythagorean theorem: [math]a^2+b^2=c^2[/math]
Descriptions
Note that when a circle equation contains a negative, h and k are actually positive values. [br][br]for the circle [math]\left(x-5\right)^2+\left(y+2\right)^2=16[/math], the center of the circle would be (h,k)=(5,-2), with a radius of 4, since[math]4^2=16[/math][br][br]Now use the geogebra applet to see how manipulating the center and radius of a circle changes the equation of a circle in real time.
Ellipse Introduction
Equation and Vocab
The standard equation of an ellipse is[br][size=100][math]\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1[/math][br][/size][br]You can see this is more involved than the circle formula, but it is related.[br][br]Ellipse Rules:[br]An ellipse is "horizontal" if the value of "a" is larger[br]An ellipse is "vertical" if the value of "b" is larger[br][br]Use the app to see how the formula changes[br]
Hyperbola Introduction
Intro
The hyperbola has two equations:[br]Horizontal: [math]\frac{\left(x-h\right)^2}{a^2}-\frac{\left(y-k\right)^2}{b^2}=1[/math][br]or [br]Vertical: [math]\frac{\left(y-k\right)^2}{b^2}-\frac{\left(x-h\right)^2}{a^2}=1[/math][br][br]If the hyperbola is horizontal, the transverse axis (the distance from the vertices) is 2a.[br]If the hyperbola is vertical, the transverse axis is 2b.[br][br]The Vertices will be:[br]Horizontal: [math]\left(h+a,k\right)\left(h-a,k\right)[/math][br]Vertical:[math]\left(h,k+b\right)\left(h,k-b\right)[/math][br][br]The The Foci Equation: [br][math]c^2=a^2+b^2[/math][br]Foci: [br]Horizontal: [math]\left(h+c,k\right)\left(h-c,k\right)[/math][br]Vertical: [math]\left(h,k+c\right)\left(h,k-c\right)[/math]