Five-Minute Check (over Lesson 7-3)

Then/Now

Key Concept:Rotation of Axes of Conics

Example 1:Write an Equation in the x′y′-Plane

Key Concept:Angle of Rotation Used to Eliminate xy-Term

Example 2:Write an Equation in Standard Form

Key Concept:Rotation of Axes of Conics

Example 3:Real World Example: Write an Equation in the xy-Plane

Example 4:Graph a Conic Using Rotations

Example 5:Graph a Conic in Standard Form

Over Lesson 7-3

A. B.

C. D.

Over Lesson 7-3

A. B.

C. D.

Over Lesson 7-3

Graph the hyperbola 4x2 – y2 + 32x + 6y + 39 = 0.

A. B.

C. D.

Over Lesson 7-3

Write an equation for the hyperbola with foci (10, –2) and (–2, –2) and transverse axis length 8.

A.

B.

C.

D.

Over Lesson 7-3

Determine the eccentricity of the hyperbola given by 9y2 – 4x2 – 18y + 24x – 63 = 0.

A. 0.555

B. 0.745

C. 1.180

D. 1.803

You identified and graphed conic sections. (Lessons 7–1 through 7–3)

• Find rotation of axes to write equations of rotated conic sections.

• Graph rotated conic sections.

Use θ = 90° to write x 2 + 3xy – y

2 = 3 in the x y -plane. Then identify the conic.

Find the equations for x and y.

Write an Equation in the x y -Plane

= –y

x = x cos θ – y sin θ Rotation equationsfor x and y

y = x sin θ + y cos θ

sin 90 = 1 and cos 90 = 0

= x

Substitute into the original equation.

x 2 + 3xy – y

2 = 3

(–y )2 + 3(–y )(x ) + (x ) 2 = 3

(y )2 – 3x y + (x ) 2 = 3

Write an Equation in the x y -Plane

Answer:

Write an Equation in the x y -Plane

Use θ = 60° to write 4x 2 + 6xy + 9y

2 = 12 in the x y -plane. Then identify the conic.

A.

B.

C.

D.

Rotation of the axes

Write an Equation in Standard Form

Using a suitable angle of rotation for the conic with equation x

2 – 4xy – 2y 2 – 6 = 0, write the equation in

standard form.

The conic is a hyperbola because B2 – 4AC > 0. Find θ.

A = 1, B = –4, and C = –2

Write an Equation in Standard Form

–3

Write an Equation in Standard Form

Use the half-angle identities to determine sin θ and cos θ.

Half-Angle Identities

Simplify.

Write an Equation in Standard Form

Next, find the equations for x and y.

Rotation equations for

x and y

Simplify.

Write an Equation in Standard Form

Substitute these values into the original equation.

x2 – 4xy – 2y2 = 6

Write an Equation in Standard Form

Answer:

A.

B.

C.

D.

Write an Equation in the xy-Plane

Use the rotation formulas for x and y to find the equation of the rotated conic in the xy-plane.

Write an Equation in the xy-Plane

Substitute these values into the original equation.

= x cos 45° + y sin 45° θ = 45° = y cos 45° – x sin 45°

Rotation equations for

x′ and y′

Write an Equation in the xy-Plane

Original equation

Multiply each side by 16.

Substitute.

Simplify.

2(x′)2 + (y′)2 = 16

Write an Equation in the xy-Plane

Answer: 3x2 + 2xy + 3y2 – 32 = 0

Combine like terms.

Simplify.

A.

B.

C.

D.

ASTRONOMY A sensor on a satellite is modeled by

after a 60° rotation. Find the equation

for the sensor in the xy-plane.

The equation represents an ellipse in standard form. Use the center (0, 0), vertices (–6, 0), (6, 0), and co-vertices (0, –3) and (0, 3) in the x′y′-plane to determine the corresponding points for the ellipse in the xy-plane.

Graph a Conic Using Rotations

Find the equations for x and y for = 60°.

Graph a Conic Using Rotations

x = x cos – y sin Rotation equations y = x sin + y cos for x and y

Use the equations to convert the xy-coordinates of the vertex into xy-coordinates.

Graph a Conic Using Rotations

= –3

Graph a Conic Using Rotations

Graph a Conic Using Rotations

The new vertices and co-vertices can be used to sketch the ellipse. They can also be used to identify the x′y′-axis.

Answer:

Graph a Conic Using Rotations

A. B.

C. D.

Use a graphing calculator to graph the conic section given by 8x2 + 5xy – 4y2 = –2.

8x2 + 5xy – 4y2 = –2 Original equation

8x2 + 5xy – 4y2 + 2 = 0 Add 2 to each side.

–4y2 + (5x)y + (8x2 + 2) = 0 y-terms inquadratic form

Graph a Conic in Standard Form

Quadratic formula

Multiply.

Graph a Conic in Standard Form

Simplify.

Graphing both equations on the same screen yields the hyperbola.

Answer:

Use a graphing calculator to graph the conic section given by 3x2 – 6xy + 8y2 + 4x – 2y = 0.

A. B.

C. D.