 The Conics

By: Diana Brown

Day Five:

Ellipses

Definition:

An ellipse is the set of all points P such that the sum of the distances between P and two distinct points, called the foci (±c, 0), is a constant.

There are two main types of ellipses:  The horizontal major axis ellipse and the vertical major axis ellipse.

Diagram of a horizontal major axis ellipse The line through the foci intersects the ellipse at two points, the vertices.  The line segment joining the vertices is the major axis, and its midpoint is the center of the ellipse.  The line perpendicular to the major axis at the center intersects the ellipse at two points called the co-vertices (0, ± b).  The line segment that joins these points is the minor axis of the ellipse.

General equation of the horizontal major axis ellipse: This ellipse has the major axis parallel to the x-axis making it open longer across.  The length is 2a.  The minor axis is parallel to the y-axis and has length 2b.  The foci points are 2c units apart.  The center of the ellipse until we translate it will remain at (0, 0).  The vertex points are at the end points of the major axis. Look at the equation.  The a value is always the biggest number.

Diagram of a vertical major axis ellipse General equation of the horizontal major axis ellipse: Notice the major axis and the minor axis have reversed.  The longer axis is now vertical.  What causes this to happen?  Look at the equation closely.  The a value is now under the y value rather than the x value in the previous diagram.  We now have an easy method to tell which way the ellipse opens.  Look to see whether the larger value is under the x value or the y value.

Summary of the characteristics of an ellipse (center at origin)

 Equation Major Axis Minor Axis Co-Vertices Horizontal (±a, 0) (0, ±b) Vertical (0, ±a) (±b, 0)

Equation for the Foci: c² = a² - b²

Example Practice Problems

1.  Sketch the graph and find the vertices, end points of the minor axis and foci points for:

x² + 4y² = 16

Solution:

First put the equation in the correct form by dividing everything by 16:

x²/16 + y²/4 = 1

The larger value is a² = 16 and b² = 4.  Since the larger value is under x, the ellipse has a horizontal major axis.

The values are a = 4, b = 2.

To find c, subtract 16 - 4 and take the square root.  Thus c = 3.5
Center at (0, 0)
Vertices:  (4, 0) and (-4, 0)
End Co-Vertices:  (0, 2) and (0, -2)
Foci:  (3.5, 0) and (-3.5, 0) 2.  Graph the ellipse given by .  Identify the center, vertices, co-vertices, foci.

Solution: You can see that:

Center: (0, 0)

The ellipse is horizontal

Vertices: (±4, 0)

Co-Vertices: (0, ±3)

a = 4 and b = 3

Major axis: 8 units long (2a)

Minor axis: 6 units long (2b)

To find the foci remember to use the formula: c² = a² - b²

c² = 4² - 3² = 16 – 9 = 7

c = The foci are located at (0, )

3.  An ellipse has its center at the origin.  Find an equation of the ellipse with Vertex (8, 0) and minor axis 4 units long.

Solution:

a = 8 and b = 2

The minor axis is 2b = 4, so b = 2.
The equation is: 4. An ellipse has its center at the origin.  Find an equation of the ellipse with vertex (0, -12) and focus ( 0, -4).

Solution

a = 12 and c = 4.

Both are on the y-axis, so the major axis is vertical.  To find b², square a and c and subtract.

b² = 144 - 16 = 128
Thus the equation is: Translation of the Ellipse The center is now at (h, k).  All values are now calculated from this point rather than from (0, 0).

Example

Sketch the graph and find the vertices, end points of the minor axis and foci for: Solution

The larger of the values is under x.  The major axis is horizontal.  a = 5, b = 4 and c = 3
Center:  (2, -1)
Vertices:  (7, -1) and (-3, -1)  (add/subtract 5 from the x-value )
End points of minor:  (2, 3) and (2, -5)  (add/subtract 4 from y-value)
Foci points:  (5, -1) and (-1, -1)  (add/subtract 3 from x-value)