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### Course: Precalculus>Unit 5

Lesson 3: Foci of an ellipse

# Foci of an ellipse from equation

Sal explains how the radii and the foci of an ellipse relate to each other, and how we can use this relationship in order to find the foci from the equation of an ellipse. Created by Sal Khan.

## Want to join the conversation?

• Has anyone found other websites/apps for practicing finding the foci of and/or graphing ellipses? It doesn't have to be as fun as this site, but anything that provided quick feedback on my answers would be useful for me.

Otherwise I will have to make up my own or buy a book. :/
• Erik-try interact math.com. Search universal -> Alg.2 -> Conic Sections - > Ellipse ...practice away.
• Can the foci ever be located along the y=axis semi-major axis (radius)?
• If the ellipse's foci are located on the semi-major axis, it will merely be elongated in the y-direction, so to answer your question, yes, they can be.
• At Sal says that the constraints make the semi-major axis along the horizontal and the semi-minor axis along the vertical. In general, is the semi-major axis always the larger of the two or is it always the x axis, regardless of size? Seems obvious but I just want to be sure. Thanks for any insight.
• The major axis is always the larger one. That's what "major" and "minor" mean -- major = larger, minor = smaller. At about , Sal points out in passing that if b > a, the vertical axis would be the major one.

"Semi-minor" and "semi-major" are used to refer to the radii (radiuses) of the ellipse. Since the radius just goes halfway across, from the center to the edge and not all the way across, it's call "semi-" major or minor (depending on whether you're talking about the one on the major or minor axis). "Semi-" means half.
• I still don't understand how d2+d1=2a. Can someone help me?
• We know foci are symmetric around the Y axis.
We picked the extreme point of d2 and d1 on a poing along the Y axis.
Since foci are at the same height relative to that point and the point is exactly in the middle in terms of X, we deduce both are the same.
Just try to look at it as a reflection around de Y axis.
• How can I find foci of Ellipse which b value is larger than a value?
Do the foci lie on the y-axis? Or do they just lie on the x-axis but have different formula to find them?
• Good question! The foci of the ellipse will aways lie on its major axis, so if you're solving for an ellipse that is taller than wide you will end up with foci on the vertical axis. I remember that Sal brings this up in one of the later videos, so you should run into it as you continue your studies.
• Are there always only two focal points in an ellipse?
• YES. The ellipse is the set of points which are at equal distance to two points (i.e. the sum of the distances) just as a circle is the set of points which are equidistant from one point (i.e. the center).
How can you visualise this? Draw an ellipse taking a string with the ends attached to two nails and a pencil. Look here for example: http://www.youtube.com/watch?v=gwMntXJdodE
• Is the foci of an ellipse at a specific point along the major axis...? How is it determined?
• yes it is. actually an ellipse is determine by its foci.
But if you want to determine the foci you can use the lengths of the major and minor axes to find its coordinates.
Lets call half the length of the major axis a and of the minor axis b. Then the distance of the foci from the centre will be equal to a^2-b^2. If the centre is on the origin u just take this distance as the x or y coordinate and the other coordinate will automatically be 0 as the foci lie either on the x or y axes. if the ellipse lies on any other point u just have to add this distance to that coordinate of the centre on which axis the foci lie.
For example let length of major axis be 10 and of the minor be 6 then u will get a & b as 5 & 3 respectively. Therefore you get the dist. of the foci from the centre as 4. If the ellipse lies on the origin the its coordinates will come out as either (4,0) or (0,4) depending on the axis. If it lies on (3,4) then the foci will either be on (7,4) or (3,8). The other foci will obviously be (-1,4) or (3,0) as the other foci will be 2x the distance between one foci and the centre. Hope this answer proves useful to you.
• Is foci the plural form of focus?
• Yes it is, just as radii is the plural of radius. Both are from Latin.
• I think it would be useful to clarify that the constant 2a relates to the diameter of the major axis which is not always parallel to the x axis as shown in the video therefore the equation could actually look like

(x-h)^2/b^2 + (y-k)^2/a^2 = 1, a>b where the major axis is perpendicular to the x axis (parallel to the y-axis)

i) For the case of the major axis of the ellipse perpendicular to the y axis (parallel to the x- axis)

(x-h)^2/a^2 + (y-k)^2/b^2 = 1, a>b

With the centre of the ellipse located at (h,k)
The foci are located at
F1(h-sqrt(a^2-b^2), k), F2(h+sqrt(a^2-b^2), k)

i) For the case of the major axis of the ellipse perpendicular to the x axis (parallel to the y- axis)

(x-h)^2/b^2 + (y-k)^2/a^2 = 1, a>b

With the centre of the ellipse located at (h,k)
The foci are located at

F1(h,k-sqrt(a^2-b^2)), F2(h,k+sqrt(a^2-b^2))

Please clarify if my interpretation is incorrect
• Your interpretation is correct. The constant 2a in the standard form equation for an ellipse with horizontal major axis (parallel to the x-axis) corresponds to the length of the major axis, while the constant 2b in the standard form equation for an ellipse with vertical major axis (parallel to the y-axis) corresponds to the length of the major axis.

Therefore, the general equation for an ellipse with center (h,k), major axis length 2a, and minor axis length 2b, can be written as:

(x-h)^2/a^2 + (y-k)^2/b^2 = 1, if the major axis is parallel to the y-axis

or

(x-h)^2/b^2 + (y-k)^2/a^2 = 1, if the major axis is parallel to the x-axis

In both cases, the foci of the ellipse are located at (h±sqrt(a^2-b^2), k) if the major axis is parallel to the y-axis, or (h, k±sqrt(a^2-b^2)) if the major axis is parallel to the x-axis.