Ellipses

The Ellipse in Standard Form

An ellipseThe set of points in a plane whose distances from two fixed points have a sum that is equal to a positive constant. is the set of points in a plane whose distances from two fixed points, called foci, have a sum that is equal to a positive constant. In other words, if points F1 and F2 are the foci (plural of focus) and d is some given positive constant then (x,y) is a point on the ellipse if d=d1+d2 as pictured below:

In addition, an ellipse can be formed by the intersection of a cone with an oblique plane that is not parallel to the side of the cone and does not intersect the base of the cone. Points on this oval shape where the distance between them is at a maximum are called verticesPoints on the ellipse that mark the endpoints of the major axis. and define the major axisThe line segment through the center of an ellipse defined by two points on the ellipse where the distance between them is a maximum.. The center of an ellipse is the midpoint between the vertices. The minor axisThe line segment through the center of an ellipse defined by two points on the ellipse where the distance between them is a minimum. is the line segment through the center of an ellipse defined by two points on the ellipse where the distance between them is at a minimum. The endpoints of the minor axis are called co-verticesPoints on the ellipse that mark the endpoints of the minor axis..

If the major axis of an ellipse is parallel to the x-axis in a rectangular coordinate plane, we say that the ellipse is horizontal. If the major axis is parallel to the y-axis, we say that the ellipse is vertical. In this section, we are only concerned with sketching these two types of ellipses. However, the ellipse has many real-world applications and further research on this rich subject is encouraged. In a rectangular coordinate plane, where the center of a horizontal ellipse is (h,k), we have

As pictured a>b where a, one-half of the length of the major axis, is called the major radiusOne-half of the length of the major axis.. And b, one-half of the length of the minor axis, is called the minor radiusOne-half of the length of the minor axis.. The equation of an ellipse in standard formThe equation of an ellipse written in the form (x−h)2a2+(y−k)2b2=1. The center is (h,k) and the larger of a and b is the major radius and the smaller is the minor radius. follows:

(x−h)2a2+(y−k)2b2=1

The vertices are (h±a,k) and (h,k±b) and the orientation depends on a and b. If a>b, then the ellipse is horizontal as shown above and if a<b, then the ellipse is vertical and b becomes the major radius. What do you think happens when a=b?

Equation	Center	a	b	Orientation
(x−1)24+(y−8)29=1	(1,8)	a=2	b=3	Vertical
(x−3)22+(y+5)216=1	(3,−5)	a=2	b=4	Vertical
(x+1)21+(y−7)28=1	(−1,7)	a=1	b=22	Vertical
x225+(y+6)210=1	(0,−6)	a=5	b=10	Horizontal

The graph of an ellipse is completely determined by its center, orientation, major radius, and minor radius, all of which can be determined from its equation written in standard from.

Example 1

Graph: (x+3)24+(y−2)225=1.

Solution:

Written in this form we can see that the center of the ellipse is (−3,2), a=4=2, and b=25=5. From the center mark points 2 units to the left and right and 5 units up and down.

Then draw an ellipse through these four points.

Answer:

As with any graph, we are interested in finding the x- and y-intercepts.

Example 2

Find the intercepts: (x+3)24+(y−2)225=1.

Solution:

To find the x-intercepts set y=0:

(x+3)24+(0−2)225=1(x+3)24+425=1(x+3)24=1−425(x+3)24=2125

At this point we extract the root by applying the square root property.

x+32=±2125x+3=±2215x=−3±2215=−15±2215

Setting x=0 and solving for y leads to complex solutions, therefore, there are no y-intercepts. This is left as an exercise.

Answer: x-intercepts: (−15±2215,0); y-intercepts: none.

Unlike a circle, standard form for an ellipse requires a 1 on one side of its equation.

Example 3

Graph and label the intercepts: (x−2)2+9(y−1)2=9.

Solution:

To obtain standard form, with 1 on the right side, divide both sides by 9.

(x−2)2+9(y−1)29=99(x−2)29+9(y−1)29=99(x−2)29+(y−1)21=1

Therefore, the center of the ellipse is (2,1), a=9=3, and b=1=1. The graph follows:

To find the intercepts we can use the standard form (x−2)29+(y−1)2=1:

x-intercepts set y=0	y-intercepts set x=0
(x−2)29+(0−1)2=1(x−2)29+1=1(x−2)2=0x−2=0x=2	(0−2)29+(y−1)2=149+(y−1)2=1(y−1)2=59y−1=±59y=1±53=3±53

Therefore the x-intercept is (2,0) and the y-intercepts are (0,3+53) and (0,3−53).

Answer:

Consider the ellipse centered at the origin,

x2+y24=1

Given this equation we can write,

(x−0)212+(y−0)222=1

In this form, it is clear that the center is (0,0), a=1, and b=2. Furthermore, if we solve for y we obtain two functions:

x2+y24=1y24=1−x2y2=4(1−x2)y=±4(1−x2)y=±21−x2

The function defined by y=21−x2 is the top half of the ellipse and the function defined by y=−21−x2 is the bottom half.

Try this! Graph: 9(x−3)2+4(y+2)2=36.

Answer:

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