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.

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.

*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?

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

What if the center isn't the origin? Is the equation still equal to one?

Regardless of where the ellipse is centered, the right hand side of the ellipse equation is always equal to 1.

Main content

Algebra (all content)

Unit 17: Lesson 5

Center and radii of an ellipse

Ellipse equation review

Google Classroom

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

What is the standard equation of an ellipse?

start fraction, left parenthesis, x, minus, h, right parenthesis, squared, divided by, a, squared, end fraction, plus, start fraction, left parenthesis, y, minus, k, right parenthesis, squared, divided by, b, squared, end fraction, equals, 1

This is the standard equation of the ellipse centered at left parenthesis, h, comma, k, right parenthesis, whose horizontal radius is a and vertical radius is b.

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

Check your understanding

Problem 1

Which ellipse is represented by the equation start fraction, left parenthesis, x, minus, 4, right parenthesis, squared, divided by, 9, end fraction, plus, start fraction, left parenthesis, y, plus, 6, right parenthesis, squared, divided by, 4, end fraction, equals, 1 ?

Want to try more problems like this? Check out this exercise.

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Ralph Turchiano
Posted 6 years ago. Direct link to Ralph Turchiano's post “Just for the sake of form...”
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)
Answer
- arora18204
  Posted 6 years ago. Direct link to arora18204's post “That would make sense, bu...”
  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.
  (12 votes)
Garima Soni
Posted 5 years ago. Direct link to Garima Soni's post “Please explain me derivat...”
Please explain me derivation of equation of ellipse.
(6 votes)
Answer
Matthew Johnson
Posted 6 years ago. Direct link to Matthew Johnson's post “*Would the radius of an e...”
*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?
(4 votes)
Answer
- Richard Smith
  Posted 6 years ago. Direct link to Richard Smith's post “I might can help with som...”
  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)
Abi
Posted 4 years ago. Direct link to Abi's post “What if the center isn't ...”
What if the center isn't the origin? Is the equation still equal to one?
(3 votes)
Answer
- Sergei N. Maderazo
  Posted 4 years ago. Direct link to Sergei N. Maderazo's post “Regardless of where the e...”
  Regardless of where the ellipse is centered, the right hand side of the ellipse equation is always equal to 1.
  (2 votes)
😊
Posted 5 years ago. Direct link to 😊's post “what isProving standard e...”
what isProving standard equation of an ellipse??
(3 votes)
Answer
Fred Haynes
Posted 3 months ago. Direct link to Fred Haynes's post “A simple question that I ...”
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)
Answer
- kubleeka
  Posted 3 months ago. Direct link to kubleeka's post “The standard equation of ...”
  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)
Fred Haynes
Posted 3 months ago. Direct link to Fred Haynes's post “This is on a different su...”
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)
Answer
Peyton
Posted 4 years ago. Direct link to Peyton's post “How do you change an elli...”
How do you change an ellipse equation written in general form to standard form. Ex: changing x^2+4y^2-2x+24y-63+0 to standard form
(1 vote)
Answer
- Osama Al-Bahrani
  Posted 4 years ago. Direct link to Osama Al-Bahrani's post “I hope this helps! https:...”
  I hope this helps! https://www.khanacademy.org/math/precalculus/conics-precalc/identifying-conic-sections-from-expanded-equations/v/identifying-conics-1
  (1 vote)
kananelomatshwele
Posted a year ago. Direct link to kananelomatshwele's post “How do I find the equatio...”
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)
Answer
- dashpointdash
  Posted 3 months ago. Direct link to dashpointdash's post “The ellipse is centered a...”
  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 :))
  (1 vote)
Dakari
Posted 4 years ago. Direct link to Dakari's post “Is there a specified equa...”
Is there a specified equation a vertical ellipse and a horizontal ellipse or should you just use the standard form of an ellipse for both?
(1 vote)
Answer
- Osama Al-Bahrani
  Posted 4 years ago. Direct link to Osama Al-Bahrani's post “For ellipses, a > b For ...”
  For ellipses, a > b
  
  For vertical ellipses, a^2 is below y^2. For horizontal ellipses, a^2 is below x^2.
  (1 vote)