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### Course: Precalculus>Unit 5

Lesson 2: Center and radii of an ellipse

# 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 would we make an equation of a tilted ellipse?
• 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.
• 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.
• One thing isn't right. My teacher taught us in school that when an ellipse is vertical the major axis is vertical as well. In similar terms, the major axis (2a) is always the bigger axis.
• Yes, you're absolutely right, where did he say that wasn't the case?
If the radius of the vertical axis is larger than that of the horizontal axis, then the y axis is the major axis, or it's considered the major radius. Also, the foci are always located on the major axis.
• 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.
• do you remember back when you did circles that what was on the other side of the equation was the radius of the circle squared? It's the same thing here, but due to the fact that there are now 2 different radii, we write them on the other side to keep them separate. You can multiply the one by whatever number you want to and it will make the ellipse bigger while keeping the same ratio of one radius to the other.