SECTION: 10 – 4 ROTATIONS

WARM-UPFind the center, vertices, foci, and the equations of the asymptotes of each hyperbola.

2 23 5

1. 116 4

x y

2 22. 4 25 8 150 121 0x y x y

3. Write the standard form of the hyperbola with vertices (–10,3) and (6,3) and foci (–12,3) and (8,3).

DISCRIMINANT. Given the equationAx2+Bxy+Cy2+Dx+Ey+F=0, the quantity B2–4AC isthe discriminant.

CLASSIFICATION OF CONIC SECTIONS BYTHE DISCRIMINANT

1. If B2–4AC<0, then the graph of the equation is either a circle or an ellipse.

2. If B2–4AC=0, then the graph of the equation is a parabola.

3. If B2–4AC>0, then the graph of the equation is a hyperbola.

ROTATED CONIC SECTIONS. Some conic sections may be rotated so that they are not parallel to either the x- or the y-axis.

GENERAL FORM OF THE EQUATION. The general form of the equation of a rotated conic section is Ax2+Bxy+Cy2+Dx+Ey+F=0. Notice the Bxy term contains both the x- and y-variables. This equation is also referred to as the equation in the xy-plane.

ROTATION OF AXES TO ELIMINATE AN xy-TERM. Rotation of the axes is the process used to eliminate the xy-term in the general form of the equation. The objective is to rotate x- and y-axes until they are parallel to the axes of the conic section. The rotated axes are denoted as the x’-axis and the y’-axis.

x

y

θ

x’

y’

FORMAL DEFINITION. The general second-degree equation Ax2+Bxy+Cy2+Dx+Ey+F=0 can be rewritten as: A’(x’)2+C’(y’)2+D’x’+E’y’+F’=0by rotating the coordinate axes through an angle θ,

where The coefficients of the new

equation are obtained by making the substitutions

cot 2 .A C

B

'cos 'sin and 'sin 'cos .x x y y x y

ELIMINATING THE xy-TERM

1. Identify the A, B, and C values.

2. Determine the angle measure of the rotation using

the formula

3. Find the value of using the half- angle formulas.

cot 2 .A C

B

sin and cos

4. Write the rotated equation by substituting into the original equation and simplify.

'cos 'sin and 'sin 'cosx x y y x y

EXAMPLE 1. Determine the type of conic section. Rotate the conic section to eliminate the xy-term. Then write the standard form of the equation.

a. 1 0xy

2 2b. 7 6 3 13 16 0x xy y

2 2c. 4 4 5 5 1 0x xy y y

d. Transform the equation to an x’y’ equation by a rotation of 45°.

2 2 4 0x xy y

CLASS WORK/HOMEWORK:

SECTION: 10 – 4

PAGE: 729

PROBLEMS: 5 – 15 ODD (DO NOT SKETCH)