Rotation of Axes: Centered Ellipse

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Analyze by rotating the axes to eliminate the term.

Let's relate this to the general second degree equation in terms of the coefficients A, B, C, D, E, F. State that equation.

The general second degree equation is

State the values of these coefficients for this problem.

State the simplified second degree equation after rotation of the coordinate axes so as to eliminate the xy term. Use A', C', D', E', F'.

Note that the term does not appear.

Why do we use ?

This equation applies to coordinates measured along the rotated axes.

We need to determine an angle, , to describe the rotation of the axes which will cause the xy term to drop out. How is that angle related to the coefficients A, B, C, D, E, F?

Determine for this example.

How can we solve this on our calculator?

We need to convert this to an expression involving the tangent so that we can use the inverse tangent function.

Do that.

Since the cotangent and tangent are reciprocals, we get

Solve for

We get woops! This is undefined.

We can try a small number, bigger than 0, in the denominator.
Set that up.

Take the inverse tangent.

We get

Is this the angle we need for the rotation of the axes?

No, we need half that, or

Next we need to determine the coefficients in the prime system. State the equation relating .

Substitute the values on the right and simplify.

We get

Find C' in a similar way.

Find the values of D' and E'.

Determine F'

Combine these results to express the original equation in terms of x' and y'.

Rewrite this as the equation of a conic section.

Dividing by 144, we get

Which conic section is this?

An ellipse.

What is the length of the semimajor axis, a?

What is the length of the semiminor axis, b?

Draw the two sets of axes, with the x-axis horizontal.

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Add the ellipse to this diagram.

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