Main content

## Precalculus (2018 edition)

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

Lesson 5: Center and radii of an ellipse# Ellipse equation review

Review your knowledge of ellipse equations and their features: center, radii, and foci.

## What is the standard equation of an ellipse?

This is the standard equation of the ellipse centered at $(h,k)$ , whose horizontal radius is $a$ and vertical radius is $b$ .

*Want to learn more about ellipse equation? Check out this video.*

## Want to join the conversation?

- Just for the sake of formality, is it better to represent the denominator (radius) as a power such as 3^2 or just as the whole number i.e. 9.(14 votes)
- That would make sense, but in a question, an equation would hardly ever be presented like that. It would make more sense of the question actually requires you to find the square root. Having 3^2 as the denominator most certainly makes sense, but it just makes the question a whole lot easier.(16 votes)

- Please explain me derivation of equation of ellipse.(7 votes)
- https://youtu.be/5TQMJ09MLWM?si=CjjRrQ661oVi685c

watch this vid it shows the derivation(1 vote)

- *Would the radius of an ellipse match the radius in the beginning of a parabola or hyperbola?* How could we calculate the area of an ellipse? Do they occur naturally in nature? Do they have any value in the real world other than mirrors and greeting cards and JS programming (https://www.khanacademy.org/computer-programming/spin-off-of-ellipse-demonstration/5350296801574912)? Can they be classified as circles? Or would circles be classified as an ellipse? How could the circumference of an ellipse be measured? Would you still use π? What about a cylinder with a base that is an ellipse instead of a circle? How would formulas change for manipulating circles vs. ellipse?(5 votes)
- I might can help with some of your questions.

Area of ellipse https://www.math.hmc.edu/funfacts/ffiles/10006.3.shtml

Ellipses occur naturally in nature - Kepler found them as descriptions of planetary orbits, and helped to begin modern astronomy. I think they also are descriptive of a cable bearing a load. Other examples can be found with a Google search

Circles are a special class of ellipse.

Ellipse perimeter is more difficult. Someone else might help.

http://mathforum.org/dr.math/faq/formulas/faq.ellipse.circumference.html(4 votes)

- What if the center isn't the origin? Is the equation still equal to one?(4 votes)
- Regardless of where the ellipse is centered, the right hand side of the ellipse equation is always equal to 1.(4 votes)

- Can someone explain to me the derivation of the Ellipse Equation? Sal explained in the intro video how it's similar to the circle, with just the radius being unequal in an ellipse, thus "a" and "b" for the denominator. However, this is still unclear to me... If someone can provide a detailed but intuitive explanation, that would be very helpful.(3 votes)
- what isProving standard equation of an ellipse??(3 votes)
- This is on a different subject. But what gives me the right to change (p-q) to (p+q) and what's it called?(1 vote)
- The algebraic rule that allows you to change (p-q) to (p+q) is called the "additive inverse property." This property states that the sum of a number and its additive inverse is always equal to zero.

In this case, the additive inverse of q is -q. Therefore, if you add -q to p-q, you get:`p - q + (-q) = p - (q + q) = p - 2q`

However, if you instead add q to both sides of p-q, you get:`p - q + q = p`

And thus, you can substitute (p+q) for (p-q) if it makes sense in the context of the problem you are trying to solve. It is important to note that this substitution is only valid if you are adding or subtracting the same value to both sides of an equation, and you should be careful to keep track of any changes you make to ensure that your answer is still correct.(2 votes)

- How do I find the equation of the ellipse with centre (0,0) on the x-axis and passing through the point (-3,2*3^2/2) and (4,4/3*5^1/2)?(1 vote)
- The ellipse is centered at (0,0) but the minor radius is uneven (-3,18?) and (4,4/3*sqrt(5)?).

We know the ellipse equation to be*x^2/a^2 + y^2/b^2 = 1*, where a is the first, b the second radius.

So - now, we have got 2 points that are satisfied by the equation.

Equation

I.:

9/a^2 + 324/b^2 = 1

II.:

16/a^2 + (32/9)/b^2 = 1

From here we solve by elimination LCM of 9 and 16 is I_.:

144/a^2 + 16*324/b^2 = 16

II_.:

144/a^2 + 9*(32/9)/b^2 = 9

I_ - II_.:

16*324/b^2 + 9*(32/9)/b^2 = 7

b^2 = 7/5216

Insert the value in II. and a^2 is -0.006

=> there is no solution

(in case I got the numbers wrong, just change them - but the procedure should be the same. Get equations and solve for radii - this all asumes math without tilted ellipses, which is a tad bit more complicated :))(2 votes)

- A simple question that I have lost sight of during my reviews of Conics. Why is the standard equation of an ellipse equal to 1?(1 vote)
- The standard equation of a circle is x²+y²=r², where r is the radius. An ellipse is a circle that's been distorted in the x- and/or y-directions, which we do by multiplying the variables by a constant. e.g. we stretch by a factor of 3 in the horizontal direction by replacing x with 3x.

If we stretch the circle, the original radius of the circle becomes irrelevant. We could just as well have started with a smaller circle and stretched it more. So we normalize the equation, and get rid of the irrelevant 'r' variable, by just dividing both sides by the original r². The right-hand side is just 1 in every case, and all the information about the stretching is encoded in just the two coefficients.

We could have chosen a different constant to make the right-hand side equal to, and it would have served the same purpose. But 1 is simplest to work with, so we use that.(1 vote)

- How do you alter the equation of an ellipse to allow for a rotation?(1 vote)