Question

What are common mistakes students make with ellipses in standard form?

Answer 1

The Standard form for an ellipse (as I teach it) looks like: `(x-h)^2/a^2+(y-k)^2/b^2=1`.
(h,k) is the center.
the distance "a" = how far right/left to move from the center to find the horizontal endpoints.
the distance "b" = how far up/down to move from the center to find the vertical endpoints.

I think that often students will mistakenly think that `a^2` is how far to move away from center to locate the endpoints. Sometimes, this would be a very large distance to travel!

Also, I think sometimes students mistakenly move up/down instead of right/left when applying these formulas to their problems.

Here is an example to talk about:
`(x-1)^2/4+(y+4)^2/9=1`

The center is (1, -4). You should move right and left "a" = 2 units to get the horizontal endpoints at (3, -4) and (-1, -4). (see image)

You should move up and down "b" = 3 units to get the vertical endpoints at (1, -1) and (1, -7). (see image)

Since a < b, the major axis is going to be in the vertical direction.
If a > b, the major axis will be going in the horizontal direction!

If you need to find out any other information about ellipses, ask another question!

Answer 2

(Confusion as to whether `a` and `b` represent the major/minor radii, or the `x`- & `y`-radii)

Recall that the standard form for an ellipse centered at the origin is

`x^2/(a^2) + y^2/b^2 = 1`

Already, however, some will take issue with the formula listed above. Some schools of thought hold that `a` should always be larger than `b` and thus represent the length of the major radius (even if the major radius lies in the vertical direction, thus allowing for `y^2/a^2` in such a case), while others hold that it should always represent the `x`-radius (even if the `x`-radius is the minor radius).

The same holds true with `b`, though in reverse. (i.e. some believe that `b` should always be the minor radius, and others believe that it should always be the `y`-radius).

Make sure you know which method your instructor (or the program you're using) prefers. If no strong preference exists, then simply decide for yourself, but be consistent with your decision. Changing your mind halfway through the assignment will make things unclear, and changing your mind halfway through a single problem will just lead to mistakes.

(Radius/axis confusion)

The majority of mistakes in ellipses seem to result from this confusion as to which radius is major and which is minor. Other possible mistakes can arise if one confuses the major radius with the major axis (or the minor radius with the minor axis). The major (or minor) axis is equal to twice the major (or minor) radius, as it is essentially the major (or minor) diameter. Depending on the step where this confusion occurs, this can lead to severe errors in scale for the ellipse.

(Radius/radius squared confusion)

A similar error occurs when students forget that the denominators (`a^2, b^2`) are the squares of the radii, and not the radii themselves. It is not uncommon to see a student with a problem such as `x^2/9 + y^2/4 = 1` draw an ellipse with `x`-radius 9 and `y`-radius 4. Further, this can occur in conjunction with the above mistake (confusing the radius for the diameter), leading to outcomes such as a student with the above equation drawing an ellipse with major diameter 9 (and thus major radius 4.5), instead of the correct major diameter 6 (and major radius 3).

(Hyperbola and Ellipse confusion) [WARNING: Answer is fairly lengthy]

Another relatively common mistake occurs if one mis-remembers the formula for the ellipse. Specifically, the most common of these errors seems to occur when one confuses the formula for ellipses with that for hyperbolas (which, recall, is `x^2/a^2 -y^2/b^2 = 1` or `y^2/b^2 - x^2/a^2 = 1` for those centered at the origin, again subject to the axis-labeling conventions listed above). For this, it helps to remember the definition of ellipses and hyperbolas as conic sections.

Specifically, recall that an ellipse is the locus of points related to two foci `f_1 & f_2` located along the major axis such that, for an arbitrary point `p` on the locus, the distance from `p` to `f_1` (labelled `d_1`) plus the distance from `p` to `f_2` (labelled `d_2`) equals twice the major radius (i.e., if `a` is the major radius, `d_1 + d_2 = 2a`). Further, the distance from the center to either of these foci (sometimes called half-focal separation or linear eccentricity), assuming `a` is the major radius, is equal to `sqrt(a^2-b^2)`.

By contrast, a hyperbola is the locus of points related to two foci in such a manner that, for a point `p` on the locus, the absolute value of the difference between the point's distance to the first focus and the point's distance to the second focus is equal to twice the major radius (i.e. with `a` major radius, `|d_1 - d_2| = 2a`). Further, the distance from the center of the hyperbola to either of these foci (again, sometimes called the linear eccentricity, and still assuming `a` major radius) is equal to `sqrt (a^2 + b^2)`.

Relating to the definition of conic sections, the overall eccentricity `e` of a section determines whether it is a circle (`e=0`), ellipse (`0 < e<1`), parabola (`e=1`), or hyperbola (`e>1`). For ellipses and hyperbolas, the eccentricity can be calculated as the ratio of the linear eccentricity to the length of the major radius; thus, for an ellipse it will be `e = sqrt (a^2-b^2) /a = sqrt (1 - b^2/a^2)` (and thus necessarily less than 1), and for a hyperbola it will be `e = sqrt(a^2+b^2)/a = sqrt (1+b^2/a^2)` (and thus necessarily greater than 1).