Programmed tutorial: Ellipse: Determine its Graph from a Description: Example 2

Find the equation of the ellipse with center at (–3, 1) and horizontal major axis 10 units long and vertical axis 4 units long.

First state the standard equation for an ellipse with a horizontal major axis.

Equation 1

What property of the ellipse is a related to?

a is half the major axis, also called the semimajor axis.

Can we determine a from the description above?

Yes.

How?

We have information on the length of the major axis to compare to a.

Do the comparison.

Major axis = 2 * semimajor axis.

Evaluate for this case and solve for a.

10 = 2 a, or a = 5.

Can we determine b from the above information?

Yes.

How?

In a way similar to what we just did for a, except we use b and length of the minor axis.

Do it.

We get 4 = 2b, or b = 2.

What property of the ellipse do h and k refer to?

h is the x-coordinate of the center and k is the y-coordinate.

Determine the values of h and k here?

Since the center is at (–3 ,1), we have h = –3 and k = 1.

Which quadrant will the center be in?

Quadrant II.

Combine our several results with Equation 1 to get the equation of this ellipse.

The equation of the ellipse is

, or

We have determined the coordinates of the center of the ellipse as (–3, 1). How far to the right of the center will the right vertex be?

The distance from the center to either vertex is a = 5.

Determine the x-coordinate of the right vertex.

The right vertex will be at x = –3 + 5 = 2.

What is the y-coordinate of this vertex?

Since the vertices and the center are all on the horizontal axis, they have the same y-coordinate.

What is that y-coordinate here?

Since k = 1, the right vertex has the coordinates (2, 1).

Analyze the left vertex in a similar way. What are its coordinates?

Its x-coordinate is –3 – 5 = –8. Its y-coordinate is 1, so the left vertex has the coordinates (–8, 1).

How high is the highest point of this ellipse?

It is a distance b above the center, which is at y = k.

What will be its y-coordinate?

y = k + b = 1 + 2 = 3.

What will its x-coordinate be?

It is the same as the x-coordinate of the center.

Combine these results to obtain the coordinates of the highest point.

The highest point will be at (–3, 3).

State the coordinates of the lowest point of this ellipse.

The y-coordinate is 1 – 2 = –1. The lowest point is at ( –3, –1).

Draw the graph of this ellipse on paper. Try to do that without using your calculator. Then check your graph by clicking “Next”.

Note that the ellipse extends ±5 units from the center.
Note the distances a and b. alt="Java applet graph of a hyperbola, foci, vertices, asymptotes." Your browser is completely ignoring the <APPLET> tag!

Next let’s add foci to the diagram. Are they inside or outside the ellipse?

Inside.

Are they on the horizontal or vertical axis?

They are on the horizontal axis.

Let c be the distance of the foci from the center. Is c larger or smaller than a?

Since the foci are on the axis, closer to the center than the vertices, we have c < a.

State the equation for c in terms of a and b.

Evaluate c for this problem.

Add the foci to your graph. Then check by clicking “Next”.

Note the distances a, b, and c. alt="Java applet graph of a hyperbola, foci, vertices, asymptotes." Your browser is completely ignoring the <APPLET> tag!

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