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## Precalculus

### Unit 5: Lesson 2

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?(26 votes)
- 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.(4 votes)

- I managed to get 5 as my major radius and 4 as my minor radius. Did I do something wrong while reading the graph?(3 votes)
- No you are correct. However the equation is written as a^2, not a.(6 votes)

- May I know where is the derivation of Ellipse equation?(3 votes)
- 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.(4 votes)

- How to chose which one is A and B in a problem in a text book?(3 votes)
- 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(2 votes)

- What is the eccentricity of an ellipse?(2 votes)
- 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.(3 votes)

- 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.(3 votes)

- 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)

## 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.