If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

## Algebra (all content)

### Unit 17: Lesson 5

Center and radii of an ellipse

# Ellipse graph from standard equation

Given the standard equation of an ellipse, Sal finds the graph of the ellipse.

## Want to join the conversation?

• hey, just asking, but in my math curriculum we had something like this, 3[x-1] squared, plus 3[y plus 3] squared. anybody understand?
(1 vote)
• You have 3(x-1)²+3(y+1)², and I assume this was set equal to a constant. I'll name the constant "3r²".
So 3(x-1)²+3(y+1)²=3r²
Divide by 3 and get (x-1)²+(y+1)²=r²
So this is the equation of a circle centered at (1, -1) with a radius of r.
(10 votes)
• hey guys an ellipse has a as length of 'radius' parallel to the horizontal axis
and b the 'radius' parallel to the vertical axis, while a hyperbola has a as whichever a or b is larger, right? it's still x^2/a^2 except a is always larger than b, right?
(3 votes)
• For an ellipse, either the a or the b can be greater. When they are equal it forms a circle.
(1 vote)
• Can you go over a question that is more complicated? I get the basics but I need more help on the intermediate stuff. Thanks!
(3 votes)
• An ellipse has an infinite number of radii like circles, or just two ?
(1 vote)
• There is no segment called the radius in an ellipse. The definition of a circle is the set of all points equidistant from a fixed point called the center (this distance is the radius). The definition of an ellipse is the set of all points such that the sum of the distances from two fixed points called the foci (plural of focus) is constant.

Imagine having two nails in a board. If you tie a string at each end to the nails and pull the string taught with your finger, you can trace an ellipse with your finger by moving it around while keeping the string taught (though you'd have to lift up the string over the nail when crossing one of the vertices). The length of the string obviously doesn't change. That length is the defining characteristic of the ellipse (along with the foci - where you place the two nails in the board).
(2 votes)
• write an equation for an ellipse if the endpoints of the major axis are at (1,6) and (1,-6) and the endpoints of the minor axis are at (5,0) and (-3,0)
(1 vote)
• How would you write an equation for ellipses if it is tilted?
(1 vote)
• To do that, you probably should do the trig thing to get the horizontal and vertical radius.
(1 vote)

## Video transcript

- [Voiceover] We're asked which ellipse is represented by the equation x minus four squared over 16 plus y minus one squared over 49 is equal to one. And we're given a bunch of choices here. We're given four choices here. So let's just think about what's going on here. So the center of the ellipse is going to be four comma one. How do I know that? Well the equation of the ellipse is going to be x minus the x-coordinate for the center squared over here, over the horizontal axis is horizontal radius squared plus y minus the y-coordinate of the center squared over the vertical radius squared. So the center is going to be four comma one. So the center here is not four comma one. The center over here is not four comma one. Not four comma one. The only choice that has a center at four comma one is this one over here. So we already know this is the choice without even looking at the horizontal and the vertical radius. But we can verify that this works out 'cause a horizontal radius right over here, notice it goes this orange line, which can represent the horizontal radius. It has a length of four and so the horizontal radius is four and so we see indeed that 16 is the horizontal radius squared, this is four squared. And if we look at the vertical radius here, we see it has a length of seven. We're going from y equals one to y equals eight, has a length of seven. And we see in that equation that this indeed is seven squared. So that was pretty straight forward.