## 17Calculus - Ellipses in Rectangular Coordinates

##### 17Calculus

An ellipse is formed when a plane intersects a cone not parallel to one of sides of the cone and not parallel to the axis of the cone. A circle is a special form of an ellipse where the plane is perpendicular to the axis of the cone.
On this page, we discuss ellipses in rectangular coordinates. For a discussion of ellipses in polar form, see this separate page.

Definition of an Ellipse

In the next section, we show two different ways to get an ellipse. There are actually three ways but we will stick with two for now. One way is to slice a cone with a plane. The other way you may be more familiar with from high school. You draw an ellipse with a piece of string and two thumbtacks.

So you may be asking why do those two different definitions define the same figure? I mean, they seem so different. Well, here is a great video that explains this. This is absolutely fascinating!

### 3Blue1Brown - Why slicing a cone gives an ellipse [12min-51secs]

video by 3Blue1Brown

Ellipse

The standard equation for an ellipse is $$\displaystyle{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 }$$. Figure 2 contains more information than we need right now but it will suffice. The longer axis is called the major axis (in this plot it is horizontal). The shorter axis is called the minor axis. The vertices are located on the ellipse where it crosses the major axis. The foci are also on the major axis, labeled F1 and F2 on this plot.

The major axis is determined by the denominators, $$a^2$$ and $$b^2$$. The larger value is in the denominator of the major axis, i.e. if $$a > b$$ then the major axis is parallel to the x-axis. We need to define a value c where $$c^2=\abs{a^2-b^2}$$ which will help us determine the location of the foci.

These tables contain the main attributes of an ellipse. We assume here that $$a > b$$. Similar equations exist for $$a < b$$.

$$\displaystyle{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 }$$

center

$$(h,k)$$

major axis

$$y=k$$

vertices

$$(h \pm a, k),$$ $$(h, k \pm b)$$

foci

$$(h \pm c,k)$$

$$c^2=\abs{a^2-b^2}$$

eccentricity

$$e=c/a$$

$$\displaystyle{ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 }$$

center

$$(h,k)$$

major axis

$$x=h$$

vertices

$$(h, k \pm a),$$ $$(h \pm b, k)$$

foci

$$(h,k \pm c)$$

$$c^2=\abs{a^2-b^2}$$

eccentricity

$$e=c/a$$

 Notes

1. Since the foci are closer to the center than the vertices, it follows that $$c < a$$ and therefore $$0 < e < 1$$.
2. Notice in the standard form of the equation, both terms are positive. This is how you know the graph is an ellipse and not a hyperbola.
3. In the general form of the equation, $$Ax^2+Bxy+Cy^2+$$ $$Dx+Ey+F=0$$, $$A > 0$$ and $$C > 0$$.
4. The eccentricity e is not the same as the irrational constant $$e \approx 2.72$$.

Okay, time for some fun videos about ellipses. Here are a couple of videos about playing pool on an elliptical table. They clearly show the relationship between the foci and demonstrate some fun physics at the same time.

### Numberphile - Elliptical Pool Table (1) [3min-39secs]

video by Numberphile

### Numberphile - Elliptical Pool Table (2) [5min-52secs]

video by Numberphile

Perimeter of an Ellipse

Did you know that there is no simple, closed form of an equation for the perimeter of an ellipse? Really?! I didn't either until recently. I've taught ellipses in precalc and calc multiple times and I never knew this! So here is the video that opened my eyes to this strange and wonderful fact. At about time index 14:11 he explains that the perimeter is actually calculated using an infinite series. Enjoy.

### Stand Up Math - Why is there no equation for the perimeter of an ellipse? [21min-4secs]

video by Stand Up Math

Practice

Write the standard form of the equation for an ellipse, centered at the origin, vertical major axis of length 8 and minor axis of length 2.

Problem Statement

Write the standard form of the equation for an ellipse, centered at the origin, vertical major axis of length 8 and minor axis of length 2.

Solution

### PatrickJMT - 1587 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Write the standard form of the equation for an ellipse, centered at the origin, with x-intercepts at $$\pm 12$$ and foci at $$(0,\pm 5)$$.

Problem Statement

Write the standard form of the equation for an ellipse, centered at the origin, with x-intercepts at $$\pm 12$$ and foci at $$(0,\pm 5)$$.

Solution

### PatrickJMT - 1588 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Write the standard form of the equation for an ellipse, centered at the origin, with minor axis of length 6 and foci at $$(\pm 8, 0)$$.

Problem Statement

Write the standard form of the equation for an ellipse, centered at the origin, with minor axis of length 6 and foci at $$(\pm 8, 0)$$.

Solution

### PatrickJMT - 1589 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Find the intercepts of the ellipse $$\displaystyle{ \frac{y^2}{100} + \frac{x^2}{121} = 1 }$$.

Problem Statement

Find the intercepts of the ellipse $$\displaystyle{ \frac{y^2}{100} + \frac{x^2}{121} = 1 }$$.

Solution

### PatrickJMT - 1590 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Graph the ellipse $$\displaystyle{ 1 - \frac{y^2}{16} = x^2 }$$.

Problem Statement

Graph the ellipse $$\displaystyle{ 1 - \frac{y^2}{16} = x^2 }$$.

Solution

### PatrickJMT - 1591 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Find the center and the radius of the circle $$x^2+2x+y^2=4$$.

Problem Statement

Find the center and the radius of the circle $$x^2+2x+y^2=4$$.

Solution

### Krista King Math - 1594 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Sketch the circle $$x^2 + y^2 = 4x$$.

Problem Statement

Sketch the circle $$x^2 + y^2 = 4x$$.

Solution

### Krista King Math - 1595 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Sketch the circle $$x^2 + y^2 + 6y = 0$$.

Problem Statement

Sketch the circle $$x^2 + y^2 + 6y = 0$$.

Solution

### Krista King Math - 1596 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Sketch the circle $$x^2 + y^2 + 2x + 2y = 2$$.

Problem Statement

Sketch the circle $$x^2 + y^2 + 2x + 2y = 2$$.

Solution

### Krista King Math - 1597 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Sketch the circle $$x^2 + y^2 + 10x - 20y + 100 = 0$$.

Problem Statement

Sketch the circle $$x^2 + y^2 + 10x - 20y + 100 = 0$$.

Solution

### Krista King Math - 1598 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Sketch the circle $$2x^2 + 2y^2 + 2x - 2y = 1$$.

Problem Statement

Sketch the circle $$2x^2 + 2y^2 + 2x - 2y = 1$$.

Solution

### Krista King Math - 1599 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Sketch the circle $$9x^2 + 9y^2 - 6x - 12y = 11$$.

Problem Statement

Sketch the circle $$9x^2 + 9y^2 - 6x - 12y = 11$$.

Solution

### Krista King Math - 1600 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Graph $$\displaystyle{ \frac{x^2}{9} + \frac{y^2}{5} = 1 }$$.

Problem Statement

Graph $$\displaystyle{ \frac{x^2}{9} + \frac{y^2}{5} = 1 }$$.

Solution

### PatrickJMT - 1601 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.