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Ellipses

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Ellipses

Study Guide

Key Definition

An ellipse is the set of all points $P$ in a plane such that the sum of the distances from $P$ to two fixed points is a given constant. These fixed points are called foci.

Important Notes

The center of the ellipse is the midpoint of the line segment joining its foci.
The major axis is the longest diameter of the ellipse.
The minor axis is perpendicular to the major axis.
The equation of an ellipse with center at the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
In the equation $b^2 = a^2 - c^2$, $c$ is the distance from the center to each focus.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\frac{a}{b}$ represents a fraction

$\sqrt{x}$ represents the square root

$x^2$ represents x squared

Remember to use proper notation when solving problems

Why It Works

Understanding the properties of ellipses helps in solving problems related to planetary orbits and optics using the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

Remember

The foci are always on the major axis, and the sum of distances from any point on the ellipse to the foci is constant.

Quick Reference

Equation:$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Major Axis:$2a$

Minor Axis:$2b$

Understanding Ellipses

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

An ellipse is shaped like an elongated circle, defined by two focal points with a constant sum of distances.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following is the equation of an ellipse centered at the origin?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine a racetrack shaped like an ellipse where the sum of distances from any point on the track to two fixed points is constant. How would you express this mathematically?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If an ellipse has its foci at $(c, 0)$ and $(-c, 0)$ with $a = 5$ and $c = 3$, find $b$.

Challenge Quiz

Single Choice Quiz

Advanced

For an ellipse centered at the origin, if $a = 7$ and $c = 4$, which of the following is $b^2$?

Recap

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Review key concepts and takeaways