Programmed tutorial: Given a Quadratic, Determine Equation of Ellipse: Example 1

Determine the standard equation of the ellipse from the general quadratic equation

General Contents

Detailed Contents

Index

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

How do we proceed to convert the given equation to the standard form?

We need to complete the squares.

What do we do with the constant?

Move it to the right-hand side.

Do that and group the terms involving the same variable.

Factor the left-hand side.

Factoring in each group, we get

To complete the square involving x, what do we need to add inside the parentheses?

We add

inside the parentheses.

For y, what do we need to add inside the parentheses?

We add

inside the parentheses.

If we make these additions inside the parentheses, what are we adding in total to the left-hand side?

Taking the multiplication into account, we are adding
(9 * 25) + (16 * 4).
As a result, we are adding 225 + 64 = 289
to the left-hand side.

Can we do this?

Only if we also add 289 to the right-hand side.

Rewrite the equation in this way.

We get

Rewrite the left-hand side in terms of squares, such as (x + 5)².

We get

In order to go further towards the standard equation of the ellipse, what value do we need to get on the right-hand side?

We need a “1” on the right.

How can we convert what we have to produce this?

We can divide both sides by 144.

Do it.

Do the cancellation on the left.

We get

Comparing this to the standard equation above, what is the value of k?

We have y – 2 = y – k, or k = 2.

In a similar way, determine the value of h.

Comparing to the standard equation, we have
x + 5 = x – h, or h = –5.

What does this tell us about the location of the center of the ellipse?

The center is located at ( – 5, 2 ).

Determine a and b for this ellipse.

We find
a² = 16, or a = 4, and
b² = 9, or b = 3

How long is the major axis?

Since a is the length of the semimajor axis, the whole major axis is 2*a = 8.

How long is the minor axis?

It is 2*b = 6.

Where is the right-hand vertex located?

It is a distance a to the right of the center.

Find the resulting x-coordinate.

The right-hand vertex is at h + a = – 5 + 4 = – 1

What is the y-coordinate of the vertex?

Since it is on the axis, the y-coordinate is k = 2.

Determine the x-coordinate of the left-hand vertex.

This vertex is at x =  h – a =  – 5 – 4 = – 9

State the coordinates of the left-hand vertex.

( –9, 2 )

Determine c, the distance from the center to a focus.

Determine the x-coordinates of the foci.

The x-coordinates are h + c and h – c, or
and , or
–5 + 2.6 = –2.4, and – 5 – 2.6 = – 7.6

Draw a graph of this ellipse on paper.
Then check your work by clicking “Next”.

Note that the vertices are the red dots and the foci are the blue dots. The distances a, b, and c are shown.
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