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Precalculus

10.4 Rotation of Axes

Precalculus10.4 Rotation of Axes
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  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section, you will:
  • Identify nondegenerate conic sections given their general form equations.
  • Use rotation of axes formulas.
  • Write equations of rotated conics in standard form.
  • Identify conics without rotating axes.

As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See Figure 1.

Figure 1 The nondegenerate conic sections

Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in Figure 2. A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.

Figure 2 Degenerate conic sections

Identifying Nondegenerate Conics in General Form

In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.

A x 2 +Bxy+C y 2 +Dx+Ey+F=0 A x 2 +Bxy+C y 2 +Dx+Ey+F=0

where A,B, A,B, and C C are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.

You may notice that the general form equation has an xy xy term that we have not seen in any of the standard form equations. As we will discuss later, the xy xy term rotates the conic whenever  B   B  is not equal to zero.

Conic Sections Example
ellipse 4 x 2 +9 y 2 =1 4 x 2 +9 y 2 =1
circle 4 x 2 +4 y 2 =1 4 x 2 +4 y 2 =1
hyperbola 4 x 2 9 y 2 =1 4 x 2 9 y 2 =1
parabola 4 x 2 =9y or 4 y 2 =9x 4 x 2 =9y or 4 y 2 =9x
one line 4x+9y=1 4x+9y=1
intersecting lines ( x4 )( y+4 )=0 ( x4 )( y+4 )=0
parallel lines ( x4 )( x9 )=0 ( x4 )( x9 )=0
a point 4 x 2 +4 y 2 =0 4 x 2 +4 y 2 =0
no graph 4 x 2 +4 y 2 =1 4 x 2 +4 y 2 =1
Table 1

General Form of Conic Sections

A conic section has the general form

A x 2 +Bxy+C y 2 +Dx+Ey+F=0 A x 2 +Bxy+C y 2 +Dx+Ey+F=0

where A,B, A,B, and C C are not all zero.

Table 2 summarizes the different conic sections where B=0, B=0, and A A and C C are nonzero real numbers. This indicates that the conic has not been rotated.

ellipse A x 2 +C y 2 +Dx+Ey+F=0, AC and AC>0 A x 2 +C y 2 +Dx+Ey+F=0, AC and AC>0
circle A x 2 +C y 2 +Dx+Ey+F=0, A=C A x 2 +C y 2 +Dx+Ey+F=0, A=C
hyperbola A x 2 C y 2 +Dx+Ey+F=0 or A x 2 +C y 2 +Dx+Ey+F=0, A x 2 C y 2 +Dx+Ey+F=0 or A x 2 +C y 2 +Dx+Ey+F=0,where A A and C C are positive
parabola A x 2 +Dx+Ey+F=0 or C y 2 +Dx+Ey+F=0 A x 2 +Dx+Ey+F=0 or C y 2 +Dx+Ey+F=0
Table 2

How To

Given the equation of a conic, identify the type of conic.

  1. Rewrite the equation in the general form, A x 2 +Bxy+C y 2 +Dx+Ey+F=0. A x 2 +Bxy+C y 2 +Dx+Ey+F=0.
  2. Identify the values of A A and C C from the general form.
    1. If A A and C C are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.
    2. If A A and C C are equal and nonzero and have the same sign, then the graph may be a circle.
    3. If A A and C C are nonzero and have opposite signs, then the graph may be a hyperbola.
    4. If either A A or C C is zero, then the graph may be a parabola.

    If B = 0, the conic section will have a vertical and/or horizontal axes. If B does not equal 0, as shown below, the conic section is rotated. Notice the phrase “may be” in the definitions. That is because the equation may not represent a conic section at all, depending on the values of A, B, C, D, E, and F. For example, the degenerate case of a circle or an ellipse is a point:
    A x 2 +By2=0, A x 2 +By2=0, when A and B have the same sign.
    The degenerate case of a hyperbola is two intersecting straight lines: A x 2 +By2=0, A x 2 +By2=0, when A and B have opposite signs.
    On the other hand, the equation, A x 2 +By2+1=0, A x 2 +By2+1=0, when A and B are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.

Example 1

Identifying a Conic from Its General Form

Identify the graph of each of the following nondegenerate conic sections.

  1. 4 x 2 9 y 2 +36x+36y125=0 4 x 2 9 y 2 +36x+36y125=0
  2. 9 y 2 +16x+36y10=0 9 y 2 +16x+36y10=0
  3. 3 x 2 +3 y 2 2x6y4=0 3 x 2 +3 y 2 2x6y4=0
  4. 25 x 2 4 y 2 +100x+16y+20=0 25 x 2 4 y 2 +100x+16y+20=0
Try It #1

Identify the graph of each of the following nondegenerate conic sections.

  1. 16 y 2 x 2 +x4y9=0 16 y 2 x 2 +x4y9=0
  2. 16 x 2 +4 y 2 +16x+49y81=0 16 x 2 +4 y 2 +16x+49y81=0

Finding a New Representation of the Given Equation after Rotating through a Given Angle

Until now, we have looked at equations of conic sections without an xy xy term, which aligns the graphs with the x- and y-axes. When we add an xy xy term, we are rotating the conic about the origin. If the x- and y-axes are rotated through an angle, say θ, θ, then every point on the plane may be thought of as having two representations: ( x,y ) ( x,y ) on the Cartesian plane with the original x-axis and y-axis, and ( x , y ) ( x , y )on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis. See Figure 3.

Figure 3 The graph of the rotated ellipse x 2 + y 2 xy15=0 x 2 + y 2 xy15=0

We will find the relationships between x x and y y on the Cartesian plane with x x and y y on the new rotated plane. See Figure 4.

Figure 4 The Cartesian plane with x- and y-axes and the resulting x′− and y′−axes formed by a rotation by an angle  θ.  θ.

The original coordinate x- and y-axes have unit vectors i i and j . j .The rotated coordinate axes have unit vectors i i and j . j . The angle θ θ is known as the angle of rotation. See Figure 5. We may write the new unit vectors in terms of the original ones.

i =cos θi+sin θj j =sin θi+cos θj i =cos θi+sin θj j =sin θi+cos θj
Figure 5 Relationship between the old and new coordinate planes.

Consider a vector u u in the new coordinate plane. It may be represented in terms of its coordinate axes.

u= x i + y j u= x (i cos θ+j sin θ)+ y (i sin θ+j cos θ) Substitute. u=ix' cos θ+jx' sin θiy' sin θ+jy' cos θ Distribute. u=ix' cos θiy' sin θ+jx' sin θ+jy' cos θ Apply commutative property. u=(x' cos θy' sin θ)i+(x' sin θ+y' cos θ)j Factor by grouping. u= x i + y j u= x (i cos θ+j sin θ)+ y (i sin θ+j cos θ) Substitute. u=ix' cos θ+jx' sin θiy' sin θ+jy' cos θ Distribute. u=ix' cos θiy' sin θ+jx' sin θ+jy' cos θ Apply commutative property. u=(x' cos θy' sin θ)i+(x' sin θ+y' cos θ)j Factor by grouping.

Because u= x i + y j , u= x i + y j , we have representations of x x and y y in terms of the new coordinate system.

x= x cos θ y sin θ and y= x sin θ+ y cos θ x= x cos θ y sin θ and y= x sin θ+ y cos θ

Equations of Rotation

If a point ( x,y ) ( x,y ) on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle θ θ from the positive x-axis, then the coordinates of the point with respect to the new axes are ( x , y ). ( x , y ). We can use the following equations of rotation to define the relationship between ( x,y ) ( x,y ) and ( x , y ): ( x , y ):

x= x cos θ y sin θ x= x cos θ y sin θ

and

y= x sin θ+ y cos θ y= x sin θ+ y cos θ

How To

Given the equation of a conic, find a new representation after rotating through an angle.

  1. Find x x and y y where x= x cos θ y sin θ x= x cos θ y sin θ and y= x sin θ+ y cos θ. y= x sin θ+ y cos θ.
  2. Substitute the expression for x x and y y into in the given equation, then simplify.
  3. Write the equations with x x and y y in standard form.

Example 2

Finding a New Representation of an Equation after Rotating through a Given Angle

Find a new representation of the equation 2 x 2 xy+2 y 2 30=0 2 x 2 xy+2 y 2 30=0 after rotating through an angle of θ=45°. θ=45°.

Writing Equations of Rotated Conics in Standard Form

Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form A x 2 +Bxy+C y 2 +Dx+Ey+F=0 A x 2 +Bxy+C y 2 +Dx+Ey+F=0 into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the x x and y y coordinate system without the x y x y term, by rotating the axes by a measure of θ θ that satisfies

cot( 2θ )= AC B cot( 2θ )= AC B

We have learned already that any conic may be represented by the second degree equation

A x 2 +Bxy+C y 2 +Dx+Ey+F=0 A x 2 +Bxy+C y 2 +Dx+Ey+F=0

where A,B, A,B, and C C are not all zero. However, if B0, B0, then we have an xy xy term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle θ θ where cot( 2θ )= AC B . cot( 2θ )= AC B .

  • If cot(2θ)>0, cot(2θ)>0, then 2θ 2θ is in the first quadrant, and θ θ is between (,45°). (,45°).
  • If cot(2θ)<0, cot(2θ)<0, then 2θ 2θ is in the second quadrant, and θ θ is between (45°,90°). (45°,90°).
  • If A=C, A=C, then θ=45°. θ=45°.

How To

Given an equation for a conic in the x y x y system, rewrite the equation without the x y x y term in terms of x x and y , y , where the x x and y y axes are rotations of the standard axes by θ θ degrees.

  1. Find cot(2θ). cot(2θ).
  2. Find sin θ sin θ and cos θ. cos θ.
  3. Substitute sin θ sin θ and cos θ cos θ into x= x cos θ y sin θ x= x cos θ y sin θ and y= x sin θ+ y cos θ. y= x sin θ+ y cos θ.
  4. Substitute the expression for x x and y y into in the given equation, and then simplify.
  5. Write the equations with x x and y y in the standard form with respect to the rotated axes.

Example 3

Rewriting an Equation with respect to the x′ and y′ axes without the x′y′ Term

Rewrite the equation 8 x 2 12xy+17 y 2 =20 8 x 2 12xy+17 y 2 =20 in the x y x y system without an x y x y term.

Try It #2

Rewrite the 13 x 2 6 3 xy+7 y 2 =16 13 x 2 6 3 xy+7 y 2 =16 in the x y x y system without the x y x y term.

Example 4

Graphing an Equation That Has No x′y′ Terms

Graph the following equation relative to the x y x y system:

x 2 +12xy4 y 2 =30 x 2 +12xy4 y 2 =30

Identifying Conics without Rotating Axes

Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is

A x 2 +Bxy+C y 2 +Dx+Ey+F=0 A x 2 +Bxy+C y 2 +Dx+Ey+F=0

If we apply the rotation formulas to this equation we get the form

A x 2 + B x y + C y 2 + D x + E y + F =0 A x 2 + B x y + C y 2 + D x + E y + F =0

It may be shown that B 2 4AC= B 2 4 A C . B 2 4AC= B 2 4 A C . The expression does not vary after rotation, so we call the expression invariant. The discriminant, B 2 4AC, B 2 4AC, is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.

Using the Discriminant to Identify a Conic

If the equation A x 2 +Bxy+C y 2 +Dx+Ey+F=0 A x 2 +Bxy+C y 2 +Dx+Ey+F=0 is transformed by rotating axes into the equation A x 2 + B x y + C y 2 + D x + E y + F =0, A x 2 + B x y + C y 2 + D x + E y + F =0, then B 2 4AC= B 2 4 A C . B 2 4AC= B 2 4 A C .

The equation A x 2 +Bxy+C y 2 +Dx+Ey+F=0 A x 2 +Bxy+C y 2 +Dx+Ey+F=0 is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.

If the discriminant, B 2 4AC, B 2 4AC, is

  • <0, <0, the conic section is an ellipse
  • =0, =0, the conic section is a parabola
  • >0, >0, the conic section is a hyperbola

Example 5

Identifying the Conic without Rotating Axes

Identify the conic for each of the following without rotating axes.

  1. 5 x 2 +2 3 xy+2 y 2 5=0 5 x 2 +2 3 xy+2 y 2 5=0
  2. 5 x 2 +2 3 xy+12 y 2 5=0 5 x 2 +2 3 xy+12 y 2 5=0
Try It #3

Identify the conic for each of the following without rotating axes.

  1. x 2 9xy+3 y 2 12=0 x 2 9xy+3 y 2 12=0
  2. 10 x 2 9xy+4 y 2 4=0 10 x 2 9xy+4 y 2 4=0

Media

Access this online resource for additional instruction and practice with conic sections and rotation of axes.

10.4 Section Exercises

Verbal

1.

What effect does the xy xy term have on the graph of a conic section?

2.

If the equation of a conic section is written in the form A x 2 +B y 2 +Cx+Dy+E=0 A x 2 +B y 2 +Cx+Dy+E=0 and AB=0, AB=0, what can we conclude?

3.

If the equation of a conic section is written in the form A x 2 +Bxy+C y 2 +Dx+Ey+F=0, A x 2 +Bxy+C y 2 +Dx+Ey+F=0, and B 2 4AC>0, B 2 4AC>0, what can we conclude?

4.

Given the equation a x 2 +4x+3 y 2 12=0, a x 2 +4x+3 y 2 12=0, what can we conclude if a>0? a>0?

5.

For the equation A x 2 +Bxy+C y 2 +Dx+Ey+F=0, A x 2 +Bxy+C y 2 +Dx+Ey+F=0, the value of θ θ that satisfies cot( 2θ )= AC B cot( 2θ )= AC B gives us what information?

Algebraic

For the following exercises, determine which conic section is represented based on the given equation.

6.

9 x 2 +4 y 2 +72x+36y500=0 9 x 2 +4 y 2 +72x+36y500=0

7.

x 2 10x+4y10=0 x 2 10x+4y10=0

8.

2 x 2 2 y 2 +4x6y2=0 2 x 2 2 y 2 +4x6y2=0

9.

4 x 2 y 2 +8x1=0 4 x 2 y 2 +8x1=0

10.

4 y 2 5x+9y+1=0 4 y 2 5x+9y+1=0

11.

2 x 2 +3 y 2 8x12y+2=0 2 x 2 +3 y 2 8x12y+2=0

12.

4 x 2 +9xy+4 y 2 36y125=0 4 x 2 +9xy+4 y 2 36y125=0

13.

3 x 2 +6xy+3 y 2 36y125=0 3 x 2 +6xy+3 y 2 36y125=0

14.

3 x 2 +3 3 xy4 y 2 +9=0 3 x 2 +3 3 xy4 y 2 +9=0

15.

2 x 2 +4 3 xy+6 y 2 6x3=0 2 x 2 +4 3 xy+6 y 2 6x3=0

16.

x 2 +4 2 xy+2 y 2 2y+1=0 x 2 +4 2 xy+2 y 2 2y+1=0

17.

8 x 2 +4 2 xy+4 y 2 10x+1=0 8 x 2 +4 2 xy+4 y 2 10x+1=0

For the following exercises, find a new representation of the given equation after rotating through the given angle.

18.

3 x 2 +xy+3 y 2 5=0,θ=45° 3 x 2 +xy+3 y 2 5=0,θ=45°

19.

4 x 2 xy+4 y 2 2=0,θ=45° 4 x 2 xy+4 y 2 2=0,θ=45°

20.

2 x 2 +8xy1=0,θ=30° 2 x 2 +8xy1=0,θ=30°

21.

2 x 2 +8xy+1=0,θ=45° 2 x 2 +8xy+1=0,θ=45°

22.

4 x 2 + 2 xy+4 y 2 +y+2=0,θ=45° 4 x 2 + 2 xy+4 y 2 +y+2=0,θ=45°

For the following exercises, determine the angle θ θ that will eliminate the xy xy term and write the corresponding equation without the xy xy term.

23.

x 2 +3 3 xy+4 y 2 +y2=0 x 2 +3 3 xy+4 y 2 +y2=0

24.

4 x 2 +2 3 xy+6 y 2 +y2=0 4 x 2 +2 3 xy+6 y 2 +y2=0

25.

9 x 2 3 3 xy+6 y 2 +4y3=0 9 x 2 3 3 xy+6 y 2 +4y3=0

26.

−3 x 2 3 xy2 y 2 x=0 −3 x 2 3 xy2 y 2 x=0

27.

16 x 2 +24xy+9 y 2 +6x6y+2=0 16 x 2 +24xy+9 y 2 +6x6y+2=0

28.

x 2 +4xy+4 y 2 +3x2=0 x 2 +4xy+4 y 2 +3x2=0

29.

x 2 +4xy+ y 2 2x+1=0 x 2 +4xy+ y 2 2x+1=0

30.

4 x 2 2 3 xy+6 y 2 1=0 4 x 2 2 3 xy+6 y 2 1=0

Graphical

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.

31.

y= x 2 ,θ= 45 y= x 2 ,θ= 45

32.

x= y 2 ,θ= 45 x= y 2 ,θ= 45

33.

x 2 4 + y 2 1 =1,θ= 45 x 2 4 + y 2 1 =1,θ= 45

34.

y 2 16 + x 2 9 =1,θ= 45 y 2 16 + x 2 9 =1,θ= 45

35.

y 2 x 2 =1,θ= 45 y 2 x 2 =1,θ= 45

36.

y= x 2 2 ,θ= 30 y= x 2 2 ,θ= 30

37.

x= ( y1 ) 2 ,θ= 30 x= ( y1 ) 2 ,θ= 30

38.

x 2 9 + y 2 4 =1,θ= 30 x 2 9 + y 2 4 =1,θ= 30

For the following exercises, graph the equation relative to the x y x y system in which the equation has no x y x y term.

39.

xy=9 xy=9

40.

x 2 +10xy+ y 2 6=0 x 2 +10xy+ y 2 6=0

41.

x 2 10xy+ y 2 24=0 x 2 10xy+ y 2 24=0

42.

4 x 2 3 3 xy+ y 2 22=0 4 x 2 3 3 xy+ y 2 22=0

43.

6 x 2 +2 3 xy+4 y 2 21=0 6 x 2 +2 3 xy+4 y 2 21=0

44.

11 x 2 +10 3 xy+ y 2 64=0 11 x 2 +10 3 xy+ y 2 64=0

45.

21 x 2 +2 3 xy+19 y 2 18=0 21 x 2 +2 3 xy+19 y 2 18=0

46.

16 x 2 +24xy+9 y 2 130x+90y=0 16 x 2 +24xy+9 y 2 130x+90y=0

47.

16 x 2 +24xy+9 y 2 60x+80y=0 16 x 2 +24xy+9 y 2 60x+80y=0

48.

13 x 2 6 3 xy+7 y 2 16=0 13 x 2 6 3 xy+7 y 2 16=0

49.

4 x 2 4xy+ y 2 8 5 x16 5 y=0 4 x 2 4xy+ y 2 8 5 x16 5 y=0

For the following exercises, determine the angle of rotation in order to eliminate the xy xy term. Then graph the new set of axes.

50.

6 x 2 5 3 xy+ y 2 +10x12y=0 6 x 2 5 3 xy+ y 2 +10x12y=0

51.

6 x 2 5xy+6 y 2 +20xy=0 6 x 2 5xy+6 y 2 +20xy=0

52.

6 x 2 8 3 xy+14 y 2 +10x3y=0 6 x 2 8 3 xy+14 y 2 +10x3y=0

53.

4 x 2 +6 3 xy+10 y 2 +20x40y=0 4 x 2 +6 3 xy+10 y 2 +20x40y=0

54.

8 x 2 +3xy+4 y 2 +2x4=0 8 x 2 +3xy+4 y 2 +2x4=0

55.

16 x 2 +24xy+9 y 2 +20x44y=0 16 x 2 +24xy+9 y 2 +20x44y=0

For the following exercises, determine the value of k k based on the given equation.

56.

Given 4 x 2 +kxy+16 y 2 +8x+24y48=0, 4 x 2 +kxy+16 y 2 +8x+24y48=0, find k k for the graph to be a parabola.

57.

Given 2 x 2 +kxy+12 y 2 +10x16y+28=0, 2 x 2 +kxy+12 y 2 +10x16y+28=0, find k k for the graph to be an ellipse.

58.

Given 3 x 2 +kxy+4 y 2 6x+20y+128=0, 3 x 2 +kxy+4 y 2 6x+20y+128=0, find k k for the graph to be a hyperbola.

59.

Given k x 2 +8xy+8 y 2 12x+16y+18=0, k x 2 +8xy+8 y 2 12x+16y+18=0, find k k for the graph to be a parabola.

60.

Given 6 x 2 +12xy+k y 2 +16x+10y+4=0, 6 x 2 +12xy+k y 2 +16x+10y+4=0, find k k for the graph to be an ellipse.

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