Subjects

Pre-Calculus

Ellipses & hyperbola

Foci of a hyperbola

Easy Guide: Solving Hyperbola Equations, Ellipse Graphs, and Parabola Secrets

Name: Knowunity
Brand: Knowunity

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Easy Guide: Solving Hyperbola Equations, Ellipse Graphs, and Parabola Secrets

emma smith

@emmasmith

1 Follower

A comprehensive guide to conic sections covering how to solve hyperbolas equations, ellipses characteristics and graphs, and parabolas focus and directrix.

Conic sections are divided into three main types: hyperbolas, ellipses, and parabolas
Each shape has unique characteristics and equations that define their behavior
Understanding orientation, vertices, and focal points is crucial for graphing
Standard forms and equations help identify and classify different conic sections
Practical problem-solving methods are provided for each type

5/22/2023

242

10.4 Hyperbolas
Definition:
The set of all (x, y)
-points in a plane, the
difference of whose
distances from two
fixed points is constant.
H

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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.

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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

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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

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Download in

Google Play

Download in

App Store

Knowunity is the # 1 ranked education app in five European countries

4.9+

Average App Rating

13 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying

Subjects

Pre-Calculus

Ellipses & hyperbola

Foci of a hyperbola

Easy Guide: Solving Hyperbola Equations, Ellipse Graphs, and Parabola Secrets

emma smith

@emmasmith

1 Follower

A comprehensive guide to conic sections covering how to solve hyperbolas equations, ellipses characteristics and graphs, and parabolas focus and directrix.

Conic sections are divided into three main types: hyperbolas, ellipses, and parabolas
Each shape has unique characteristics and equations that define their behavior
Understanding orientation, vertices, and focal points is crucial for graphing
Standard forms and equations help identify and classify different conic sections
Practical problem-solving methods are provided for each type

5/22/2023

242

11th

Pre-Calculus

Page 1: Hyperbolas

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.

Page 2: Ellipses

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

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Download in

Google Play

Download in

App Store

4.9+

Average App Rating

13 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User