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## NASA

### Course: NASA>Unit 2

Lesson 3: Orbital mechanics

# Intro to ellipses

Learn all about ellipses in this video. The standard form for an ellipse centered at the origin is x²/a² + y²/b² = 1. The semi-major axis is the longest radius and the semi-minor axis is the shortest radius. The video also explains how to shift an ellipse. Created by Sal Khan and NASA.

## Want to join the conversation?

• Why do ellipses and hyperbolas equal one?
• Because they are 'deformations' of a circle. You know what an equation of a circle looks like, right? Suppose you have this one:

(x - 3)² + (y - 2)² = 4

The center is at (3, 2), but how can one squeeze the circle to make it appear as an ellipse or hyperbola? Divide both side by 4 and you get:

(x - 3)²/4 + (y - 2)²/4 = 1

It's still the same circle, but now you know where that 1 is coming from and you can also squeeze it by changing the denominator of either the x or y term. In this video you can see what happens when the denominator of one of the terms changes:

https://www.screenr.com/lBZN

The equation 'd' is the one I've written above and equation 'e' is:

(x - 3)²/4 + (y - 2)²/b = 1

Where b is the variable that we're changing. Notice that when b = 4, it forms the same circle as 'd', but when b =/ 4 and still positive it's an ellipse. When it goes to negative, it becomes a hyperbola.
• how can we recognize what conic section it is when the equation is not written in standard form?
• For a parabola: there is only one squared term in the equation
Circle: the coefficents are the same for the two squared terms
Ellipses: the two fractions are added together
Hyperbola: the two fractions are subtracted from each other
• What is the difference between an oval and an ellipse?
• The term "oval" isn't really used much in geometry because it does not have a very clear definition. Thus, you'd have to ask for clarification if someone mentioned an oval in geometry.

Since oval does not have a standard exact meaning in mathematics, we cannot really compare it to an ellipse which does have a clear meaning.
• How would we make an equation of a tilted ellipse?
• Why did he subtract 5 when moving in the positive direction and add 2 while moving in the negative direction?
• In the original equation, letting x=0 gives us a certain point. We want to shift the figure to the right by 5. That is, we want x=5 in the new equation to give the same point as x=0 in the original. So we replace x with x-5, because plugging in 5 to this equation gives 0.
• Is the relationship between Circles and Ellipses similar to the relationship between Squares and Rectangles?
• Yes. A square is a type of rectangle, and a rectangle is a square after a stretch or compression. A circle is a type of ellipse, and an ellipse is a circle after a stretch or compression.
• What is the official definition of an ellipse?
• An ellipse is the set of all points the sum of whose distances from two fixed points is constant. The two fixed points are called the foci (each is a focus), and the sum of distances to the foci is the diameter of the ellipse.
• So conic sections aren't functions, right?
(1 vote)
• Some conics can be defined as a function. (1 to 1)
The majority are better described in the parametric form.