## 17Calculus - Ellipses in Rectangular Coordinates

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.

 Figure 1 Figure 2 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

### 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

### 1587 video

video by PatrickJMT

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

### 1588 video

video by PatrickJMT

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

### 1589 video

video by PatrickJMT

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

### 1590 video

video by PatrickJMT

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

### 1591 video

video by PatrickJMT

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

### 1594 video

video by Krista King Math

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

Problem Statement

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

Solution

### 1595 video

video by Krista King Math

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

Problem Statement

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

Solution

### 1596 video

video by Krista King Math

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

Problem Statement

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

Solution

### 1597 video

video by Krista King Math

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

### 1598 video

video by Krista King Math

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

Problem Statement

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

Solution

### 1599 video

video by Krista King Math

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

Problem Statement

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

Solution

### 1600 video

video by Krista King Math

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

### 1601 video

video by PatrickJMT

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