Rotating Conic Sections

Dr. ShildneckFall, 2014

Rotating Conic Sections• By imposing a “new” coordinate system that

rotates the x- and y-axes we can more easily graph the equation of a rotated conic.

• To do so, we must first determine the angle of rotation for the given equation.

• To find the angle of rotation, solve the equation for θ:

cot(2 )A C

B

Once we have determined the angle of rotation, convert the x’s and y’s in the original equation into “new” x’s and y’s using the following equations:

The new coordinate system will consist of theX’-axis and the y’-axis.

'cos 'sin

'sin 'cos

x x y

y x y

Once you have simplified the “new” versions of x and y,

1. Substitute the expressions into the original equation. (This eliminates the xy-term)

2. Utilize appropriate methods, such as completing the square, to put in standard form. (This is standard form on the rotated coordinate plane)

3. Use the new equation to plot the graph onto the new coordinate plane.

Classifying ConicsIf the graph of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is a conic, then the type of conic can be determined as follows:

Discriminant Type of Conic

B2 – 4AC < 0, and A = C Circle

B2 – 4AC < 0, and A ≠ C, (or if B ≠ 0) Ellipse

B2 – 4AC = 0 Parabola

B2 – 4AC > 0 Hyperbola

If B = 0, each axis of the conic section is horizontal or vertical.

If B ≠ 0, the axes of the conic are rotated (not horizontal/vertical).

30 or 6

2 27 6 3 13 64 0x xy y Classify and Graph the conic section

Writing the Equation of a Rotated Conic Section (for Project)

1. Determine an appropriate angle of rotation for your design. It would probably be easiest if it was a multiple of 30 or 45 degrees.

2. Draw the x’ and y’ axes in order to place your conic. Remember that the origin does not change!

3. Write the Standard Equation for your conic in terms of x’ and y’.

4. Now, convert the x’ and y’ in your equation into x and y using the following equations:

' cos sin

' sin cos

x x y

y x y

5. Substitute the expressions into the your equation for x’ and y’.

6. Expand the expressions, simplify and put in the General Form. (Expand the squares, multiply through by the LCD, and get all terms on one side - set equal to zero.)

Write the equation of the conic section

'x'y

END

• This is the end of the notes for today…