Page 2: Ellipses

This page covers ellipses, their definition, and problem-solving approaches. The content focuses on understanding the relationship between foci and the constant sum of distances that characterizes an ellipse.

Definition: An ellipse is the set of all points in an x-y plane where the sum of distances from two fixed points (foci) remains constant.

Highlight: Problem-solving for ellipses involves three main steps: determining orientation, analyzing characteristics/graphs in relation to equations, and finding key points (h,k,a,b,c).

Example: For a vertical ellipse with vertex (0,8), focus (0,4), and center (0,0):

• Uses the standard form $y-k$²/a² + $x-h$²/b² = 1
• Results in equation $y-0$²/64 + $x-0$²/16 = 1

Page 3: Parabolas

This page details parabolas, their definition, and practical applications. It emphasizes the relationship between the focus and directrix in forming a parabola.

Definition: A parabola is the set of all points in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.

Highlight: Key characteristics include:

• Directrix and focus have opposite orientations
• Focus is always inside the parabola's curve

Example: For a problem with vertex at (0,0):

• Uses the equation $y-0$² = 4p$x-0$
• Results in p = -9/2

Vocabulary:

• Directrix: The fixed line from which points on the parabola are measured
• Focus: The fixed point from which points on the parabola are measured

Page 1: Hyperbolas

This page introduces hyperbolas and their fundamental properties. A hyperbola is defined by points whose difference in distances from two fixed points remains constant. The page outlines a systematic approach to solving hyperbola problems and graphing them accurately.

Definition: A hyperbola is the set of all (x,y) points in a plane where the difference of distances from two fixed points is constant.

Highlight: To graph a hyperbola, follow these steps: determine orientation, locate center and vertices, graph conjugate axis, establish boundaries, draw asymptotes, and sketch branches.

Example: For the equation y²/9 - x²/25 = 1:

• Center at (0,0)
• Vertices at (0, ±3)
• Foci at (0, ±√34)

Vocabulary: Asymptotes are the diagonal lines that the hyperbola branches approach but never touch.