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.

## Precalculus (2018 edition)

### Course: Precalculus (2018 edition)>Unit 2

Lesson 5: Center and radii of an ellipse

# Ellipse standard equation from graph

Given an ellipse on the coordinate plane, Sal finds its standard equation, which is an equation in the form (x-h)²/a²+(y-k)²/b²=1.

## Want to join the conversation?

• Why is the equation always equal to 1?
• Let's write the circle equation:
x^2+y^2=r^2
Let's divide both sides by r^2, we get
x^2/r^2 + y^2/r^2 = r^2/r^2
r^2/r^2= 1
That is our ellipse equation. This way we can conclude that circle is a special case of ellipse, as the radius are equal. a^2 and b^2 of ellipse equation just means that there are two radius. Think of it this way a(or any other variable)= radiusX and b= radiusY. Also x^2/radiusX means the radius which extends in x direction and y^2/radiusY is the radius in y direction on the x,y plane
Sal explained it in one of the videos.
• I managed to get 5 as my major radius and 4 as my minor radius. Did I do something wrong while reading the graph?
• No you are correct. However the equation is written as a^2, not a.
• May I know where is the derivation of Ellipse equation?
• Here is the explanation:
We know, the circle is a special case of ellipse. The standard equation for circle is x^2 + y^2 = r^2
Now divide both sides by r and you will get
x^2/r^2 + y^/r^2 = 1.
Now, in an ellipse, we know that there are two types of radii, i.e. , let say a (semi-major axis) and b(semi-minor axis), so the above equation will reduce to x^2/a^2 + y^2/b^2 = 1, which is the equation of ellipse.
Again , if semi-minor axis will be equal to semi-major axis (a=b=r, say r is radius of circle), then the ellipse will again become circle and equation for ellipse will again reduce to x^2 + y^2 = r^2.
Hope it will make you understand.
Thank You.
• How to chose which one is A and B in a problem in a text book?
• it is upto you if you want a>b in any case then take a as the larger value whereas if you want a to be the length of the axis parallel to x-axis the take it to be the denominator of x but in that case for a vertical ellipse b>a. Depends on the way you write the equation
• What is the eccentricity of an ellipse?
• I don't think Sal mentions eccentricity in this video. You don't need it, here. If you're looking for a definition, you can find it easily in Wikipedia, under Ellipse.
• Hi may someone help me out. Would this equation be considered to represent an ellipse?
x^2/1 + y^2/121 = 1

And confirm the only difference between the standard form of an ellipse and a hyperbola is that the equation is finding the difference for the hyperbola and the sum for the ellipse. This is just an extra question. My biggest concern is my first question. Thank you.
(1 vote)
• Yes, this is an ellipse with major radius 11, minor radius 1, centered at the origin.
• I am just wondering why x (or y) value square must be divided by radius square in the equation of ellipse? And why is it always equal to 1?
(1 vote)
• The answer is in your question. If you divide both sides by r^2 you get x^2/r^2 + y^2/r^2 = r^2/r^2
now r^2/r^2 just equals 1.
So we have x^2/r^2 + y^2/r^2 = 1
So why is circle a special ellipse? well cause both the radii are equal. In an ellipse both take different values i.e a and b (or any other variable)
(1 vote)
• Is the variable a always the horizontal line, or can it be the vertical line? My teacher told me it can be both but this video tells me it is strictly the horizontal line.
(1 vote)
• It can whatever you like, so long as you make it clear at the beginning which variable you're using for what.
(1 vote)
• Can we get another video involving how to solve for both types of radii? I would also appreciate a way to know which part of an ellipse equation stands for what.
(1 vote)
• I'm trying to draw and ellipse around a line of best fit. Do i just surround the data with an ellipse or is there an equation or method i should use to be more accurate?
(1 vote)