Main content

## Center and radii of an ellipse

# Ellipse standard equation from graph

## Video transcript

- [Voiceover] We have an
ellipse graphed right over here. What we're going to try to do is find the equation for this ellipse. Like always, pause this video and see if you can figure it out on your own. All right, let's just remind ourselves the form of an equation of an ellipse. Let's say our ellipse is centered at the point ... I'm going to speak in generalities first and then we'll think
about the specific numbers for this particular ellipse. Say the center is at the point H,K and let's say that you
have a horizontal radius. So the radius in the X direction, horizontal radius, radius is equal to A. Let's say your vertical radius, let's say your vertical radius, radius is equal to B. Then the equation of this ellipse is going to be, is going to be X - H, X - H squared over your horizontal radius squared, so your radius in the X direction squared, plus, plus, now we'll think
about what we're doing in the vertical direction. Y - , Y - the Y coordinate of our center, so Y - K squared, over the vertical radius squared, B squared is equal to 1. Is equal to 1. What are H and K and A
and B in this situation? Well, H and K are pretty
easy to figure out. The center of this ellipse
is at the point ... See the X coordinate is -4 and the Y coordinate is 3. So this right over here is -4 and this right over here is positive 3. What is A going to be? A is your horizontal radius, your radius in the horizontal direction, so it's the length of
this line right over here. We can see it's 1, 2, 3, 4, 5 units long. A in this case is equal to 5. This is going to be 5 squared. And B is our radius in
the vertical direction. We can see it's 1, 2, 3, 4 units. So B is equal to 4. So that is 4. We can rewrite this as, we can rewrite this as X - -4, and we can simplify that in a second. X - -4 squared over 5 squared over our horizontal radius squared, so it's going to be 25 plus Y - 3 squared. Y minus the Y coordinate of our center. Y - 3 squared over our vertical radius squared, so B squared is going to be 16, and that is going to be equal to 1. Of course we could
simplify this a little bit. If I subtract a negative, that's the same thing
as adding a positive, so I can get rid of ... Instead of saying X - -4, I could just say X + 4. There you have it. We have the equation for this ellipse.