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Ellipse foci review

Review your knowledge of the foci of an ellipse.

What are the foci of an ellipse?

The start color #ed5fa6, start text, f, o, c, i, end text, end color #ed5fa6 of an ellipse are two points whose sum of distances from any point on the ellipse is always the same. They lie on the ellipse's start color #1fab54, start text, m, a, j, o, r, space, r, a, d, i, u, s, end text, end color #1fab54.
The distance between each focus and the center is called the focal length of the ellipse. The following equation relates the focal length f with the major radius p and the minor radius q:
f, squared, equals, p, squared, minus, q, squared
Want to learn more about the foci of an ellipse? Check out this video.

Finding the foci of an ellipse

Given the radii of an ellipse, we can use the equation f, squared, equals, p, squared, minus, q, squared to find its focal length. Then, the foci will lie on the major axis, f units away from the center (in each direction). Let's find, for example, the foci of this ellipse:
We can see that the major radius of our ellipse is 5 units, and its minor radius is 4 units.
f2=p2q2f2=5242f2=9f=3\begin{aligned} f^2&=p^2-q^2 \\\\ f^2&=5^2-4^2 \\\\ f^2&=9 \\\\ f&=3 \end{aligned}
The major axis is the horizontal one, so the foci lie start color #1fab54, 3, end color #1fab54 units to the right and left of the center. In other words, the foci lie at left parenthesis, minus, 4, plus minus, start color #1fab54, 3, end color #1fab54, comma, 3, right parenthesis, which are left parenthesis, minus, 7, comma, 3, right parenthesis and left parenthesis, minus, 1, comma, 3, right parenthesis.

Check your understanding

Problem 1
Plot the foci of this ellipse.

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

Want to join the conversation?

  • hopper cool style avatar for user 🤔 ᴄᴏᴅᴇᴅ ɢᴇɴɪᴜȿ 😎
    Are co-vertexes just the y-axis minor or major radii? Or is it always the minor radii either x or y-axis?
    (13 votes)
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    • blobby green style avatar for user Andrew
      co-vertices are always the endpoints of the minor axis.

      For an ellipse centered at the origin:
      -if the minor axis is horizontal, then the co-vertices will be the 2 x-intercepts
      -if the minor axis is vertical, then the co-vertices will be the 2 y-intercepts
      (3 votes)
  • leaf green style avatar for user cooper finnigan
    Does the sum of the two distances from a point to its focus always equal 2*major radius, or can it sometimes equal something else?
    (2 votes)
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    • blobby green style avatar for user Andrew
      Yes, they always equals twice the major radius. Think about this: if you take a piece of wood, and draw a line on it, then stick a nail at each of the line. Now take a piece of string, and tie each end of the string to one of the nails. The string needs to be longer than the line you drew, so that it's loose with no tension. Then you can take a pen/pencil, use it to push up against the string, and then the string can act as a guide for the pen, and you can trace out an ellipse. The two nails are the foci, and the length of the string is 2*major radius: it's a constant, and never changes.
      (19 votes)
  • marcimus pink style avatar for user Sarafanjum
    How was the foci discovered?
    (4 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      The first mention of "foci" was in the multivolume work Conics by the Greek mathematician Apollonius, who lived from c. 262 - 190 BCE.

      One theory is that the Ancient Greeks began studying these shapes - ellipses, parabolas, hyperbolas - as they were using sundials to study the sun's apparent movement.
      (4 votes)
  • blobby green style avatar for user broadbearb
    cant the foci points be on the minor radius as well? Seems like it would work exactly the same
    (4 votes)
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    • hopper cool style avatar for user obiwan kenobi
      In an ellipse, foci points have a special significance. Any ray emitted from one focus will always reach the other focus after bouncing off the edge of the ellipse (This is why whispering galleries are in the shape of an ellipsoid). The foci can only do this if they are located on the major axis.
      (1 vote)
  • blobby purple style avatar for user Amy Yu
    The equations of circle, ellipse, parabola or hyperbola are just equations and not function right? Is it because when y is squared, the function cannot be defined?
    (2 votes)
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  • blobby green style avatar for user elagolinea
    How do I get the directrix?
    (1 vote)
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  • aqualine ultimate style avatar for user Muinuddin Ahmmed
    What is the eccentricity of an ellipse?
    (1 vote)
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    • leaf green style avatar for user kubleeka
      Eccentricity is a measure of how close the ellipse is to being a perfect circle. A perfect circle has eccentricity 0, and the eccentricity approaches 1 as the ellipse stretches out, with a parabola having eccentricity exactly 1.

      You can compute the eccentricity as c/a, where c is the distance from the center to a focus, and a is the length of the semimajor axis.
      (5 votes)
  • aqualine ultimate style avatar for user Yves
    Why aren't there lessons for finding the latera recta and the directrices of an ellipse?
    (2 votes)
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  • piceratops seedling style avatar for user Herdy
    How do I find the length of major and minor axis? Didn't quite understand.
    (1 vote)
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    • leaf orange style avatar for user D. v.
      There's no difficulty to find them. Simply start from the center of the ellipsis, then follow the horizontal or vertical direction, whichever is the longest, until your encounter the vertex.

      Did you mean "the foci positions" or something else ?
      (3 votes)
  • blobby green style avatar for user Fred Haynes
    A question about the ellipse at the very top of the page. How is the focus in pink the same length as each other? Clearly, there is a much shorter line and there is a longer line.
    (1 vote)
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