Question 1

Select the graph of degenerate conic.​ $$x ^ { 2 } + 2 x y + y ^ { 2 } - 9 = 0$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 2

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $48 x ^ { 2 } + 69 x y + 25 y ^ { 2 } = 50$ ​

Accepted Answer

A)  $\theta = 78.87 ^ { \circ }$ 
B)  $\theta = 77.87 ^ { \circ }$ 
C)  $\theta = 79.87 ^ { \circ }$ 
D)  $\theta = 81.87 ^ { \circ }$ 
E)  $\theta = 80.87 ^ { \circ }$ 
A)  $\theta = 78.87 ^ { \circ }$ 
B)  $\theta = 77.87 ^ { \circ }$ 
C)  $\theta = 79.87 ^ { \circ }$ 
D)  $\theta = 81.87 ^ { \circ }$ 
E)  $\theta = 80.87 ^ { \circ }$

Question 3

Use the discriminant to classify the graph.​ $x ^ { 2 } + x y + 10 y ^ { 2 } + x + y - 10 = 0$ ​

Accepted Answer

A) The graph is a cone. 
B) The graph is a parabola. 
C) The graph is a ellipse or circle. 
D) The graph is a hyperbola. 
E) The graph is a line. 
A) The graph is a cone. 
B) The graph is a parabola. 
C) The graph is a ellipse or circle. 
D) The graph is a hyperbola. 
E) The graph is a line.

Question 4

The $x ^ { \prime } y ^ { \prime }$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system.The coordinates of a point in the $x y$ -coordinate system are given.Find the coordinates of the point in the rotated coordinate system.​ $\theta = 30 ^ { \circ }$ , $( 2,6 )$ ​

Accepted Answer

A)  $( \sqrt { 3 } + 3,3 \sqrt { 3 } + 1 )$ 
B)  $( \sqrt { 3 } + 3,3 \sqrt { 3 } - 1 )$ 
C)  $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 1 } { 2 } \right)$ 
D)  $( \sqrt { 3 } - 3,3 \sqrt { 3 } - 1 )$ 
E)  $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 3 } { 2 } \right)$ 
A)  $( \sqrt { 3 } + 3,3 \sqrt { 3 } + 1 )$ 
B)  $( \sqrt { 3 } + 3,3 \sqrt { 3 } - 1 )$ 
C)  $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 1 } { 2 } \right)$ 
D)  $( \sqrt { 3 } - 3,3 \sqrt { 3 } - 1 )$ 
E)  $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 3 } { 2 } \right)$

Question 5

Consider the equation.​ $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ ​ Without calculating,explain how to rewrite the equation so that it does not have an $$x y$$ -term.
​

Accepted Answer

A) To rewrite the equation $$6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$$ so that it does not have an $x y$ -term,you can solve for $y$ in terms of $x$ by completing the square or using the Quadratic Formula. 
B) To rewrite the equation $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ so that it does not have an $x y$ -term,you can solve for $x$ in terms of $y$ by completing the square or using the Quadratic Formula. 
C) To rewrite the equation $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ so that it has an $x y$ -term,you can solve for $y$ $y$ in terms of $x$ by taking square root or using the Quadratic Formula. 
D) To rewrite the equation $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ so that it has an $x y$ -term,you can solve for $x$ in terms of $y$ by completing the square or using the Quadratic Formula. 
E) To rewrite the equation $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ so that it has an $x y$ -term,you can solve for $y$ in terms of $x$ by completing the square or using the Quadratic Formula. 
A) To rewrite the equation $$6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$$ so that it does not have an $x y$ -term,you can solve for $y$ in terms of $x$ by completing the square or using the Quadratic Formula. 
B) To rewrite the equation $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ so that it does not have an $x y$ -term,you can solve for $x$ in terms of $y$ by completing the square or using the Quadratic Formula. 
C) To rewrite the equation $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ so that it has an $x y$ -term,you can solve for $y$ $y$ in terms of $x$ by taking square root or using the Quadratic Formula. 
D) To rewrite the equation $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ so that it has an $x y$ -term,you can solve for $x$ in terms of $y$ by completing the square or using the Quadratic Formula. 
E) To rewrite the equation $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ so that it has an $x y$ -term,you can solve for $y$ in terms of $x$ by completing the square or using the Quadratic Formula.

Question 6

Consider the equation.​ $10 x ^ { 2 } - 5 x y + 10 y ^ { 2 } - 81 = 0$ ​ Explain how to identify the graph of the equation.
​

Accepted Answer

A) Find the discriminant of the equation to identify the graph of the equation. 
B) Find the foci of the equation to identify the graph of the equation. 
C) Find the major axis of the equation to identify the graph of the equation. 
D) Find the minor axis of the equation to identify the graph of the equation. 
E) Find the vertices of the equation to identify the graph of the equation. 
A) Find the discriminant of the equation to identify the graph of the equation. 
B) Find the foci of the equation to identify the graph of the equation. 
C) Find the major axis of the equation to identify the graph of the equation. 
D) Find the minor axis of the equation to identify the graph of the equation. 
E) Find the vertices of the equation to identify the graph of the equation.

Question 7

Identify the conic by writing the equation in standard form. $8 x ^ { 2 } + 8 y ^ { 2 } + 32 x + 144 y + 520 = 0$

Accepted Answer

A)  $$\frac { ( x + 2 ) ^ { 2 } } { \frac { 69 } { 8 } } + \frac { ( y + 9 ) ^ { 2 } } { \frac { 69 } { 8 } } = 1$$ ;ellipse 
B)  $\frac { ( x + 2 ) ^ { 2 } } { \frac { 11 } { 8 } } + \frac { ( y + 9 ) ^ { 2 } } { \frac { 11 } { 8 } } = 1$ ;ellipse 
C)  $( 8 x + 2 ) ^ { 2 } + ( 8 y + 9 ) ^ { 2 } = 320$ ;circle 
D)  $( x + 2 ) ^ { 2 } + ( y + 9 ) ^ { 2 } = 40$ ;circle 
E)  $( x + 2 ) ^ { 2 } + ( y + 9 ) ^ { 2 } = 20$ ;circle 
A)  $$\frac { ( x + 2 ) ^ { 2 } } { \frac { 69 } { 8 } } + \frac { ( y + 9 ) ^ { 2 } } { \frac { 69 } { 8 } } = 1$$ ;ellipse 
B)  $\frac { ( x + 2 ) ^ { 2 } } { \frac { 11 } { 8 } } + \frac { ( y + 9 ) ^ { 2 } } { \frac { 11 } { 8 } } = 1$ ;ellipse 
C)  $( 8 x + 2 ) ^ { 2 } + ( 8 y + 9 ) ^ { 2 } = 320$ ;circle 
D)  $( x + 2 ) ^ { 2 } + ( y + 9 ) ^ { 2 } = 40$ ;circle 
E)  $( x + 2 ) ^ { 2 } + ( y + 9 ) ^ { 2 } = 20$ ;circle

Question 8

Use the discriminant to classify the graph.​ $x ^ { 2 } - 10 x y - 9 y ^ { 2 } + 8 x - 33 = 0$ ​

Accepted Answer

A) The graph is a ellipse. 
B) The graph is a circle. 
C) The graph is a hyperbola. 
D) The graph is a line. 
E) The graph is a parabola. 
A) The graph is a ellipse. 
B) The graph is a circle. 
C) The graph is a hyperbola. 
D) The graph is a line. 
E) The graph is a parabola.

Question 9

Select the graph of the following equation,showing both sets of axes.​ $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 2 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 2 } = 1$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 10

Select the graph of degenerate conic.​ $$9 x ^ { 2 } - 2 x y + 9 y ^ { 2 } = 0$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 11

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $5 x ^ { 2 } - 4 x y + 6 y ^ { 2 } = 9$ ​

Accepted Answer

A)  $\theta = 79.87 ^ { \circ }$ 
B)  $\theta = 77.87 ^ { \circ }$ 
C)  $\theta = 81.87 ^ { \circ }$ 
D)  $\theta = 80.87 ^ { \circ }$ 
E)  $\theta = 78.87 ^ { \circ }$ 
A)  $\theta = 79.87 ^ { \circ }$ 
B)  $\theta = 77.87 ^ { \circ }$ 
C)  $\theta = 81.87 ^ { \circ }$ 
D)  $\theta = 80.87 ^ { \circ }$ 
E)  $\theta = 78.87 ^ { \circ }$

Question 12

Use the Quadratic Formula to solve for $y$ .​ $x ^ { 2 } - 5 x y - 4 y ^ { 2 } + 3 x - 32 = 0$ ​

Accepted Answer

A)  $y = \frac { - 5 x \pm \sqrt { 25 x ^ { 2 } + 16 \left( x ^ { 2 } - 3 x - 32 \right) } } { 8 }$ 
B)  $y = \frac { - 5 x \pm \sqrt { 25 x ^ { 2 } - 16 \left( x ^ { 2 } + 3 x - 32 \right) } } { 8 }$ 
C)  $y = \frac { - 5 x \pm \sqrt { 25 x ^ { 2 } - 16 \left( x ^ { 2 } + 3 x + 32 \right) } } { 8 }$ 
D)  $y = \frac { - 5 x \pm \sqrt { 25 x ^ { 2 } + 16 \left( x ^ { 2 } + 3 x - 32 \right) } } { 8 }$ 
E)  $y = \frac { 5 x \pm \sqrt { 25 x ^ { 2 } + 16 \left( x ^ { 2 } + 3 x - 32 \right) } } { 8 }$ 
A)  $y = \frac { - 5 x \pm \sqrt { 25 x ^ { 2 } + 16 \left( x ^ { 2 } - 3 x - 32 \right) } } { 8 }$ 
B)  $y = \frac { - 5 x \pm \sqrt { 25 x ^ { 2 } - 16 \left( x ^ { 2 } + 3 x - 32 \right) } } { 8 }$ 
C)  $y = \frac { - 5 x \pm \sqrt { 25 x ^ { 2 } - 16 \left( x ^ { 2 } + 3 x + 32 \right) } } { 8 }$ 
D)  $y = \frac { - 5 x \pm \sqrt { 25 x ^ { 2 } + 16 \left( x ^ { 2 } + 3 x - 32 \right) } } { 8 }$ 
E)  $y = \frac { 5 x \pm \sqrt { 25 x ^ { 2 } + 16 \left( x ^ { 2 } + 3 x - 32 \right) } } { 8 }$

Exam 63: Rotation of Conics

Select the graph of degenerate conic.​ x2+2xy+y2−9=0x ^ { 2 } + 2 x y + y ^ { 2 } - 9 = 0x2+2xy+y2−9=0 ​

Determine the angle θ\thetaθ through which the axes are rotated.Round your answer to two decimal places.​ 48x2+69xy+25y2=5048 x ^ { 2 } + 69 x y + 25 y ^ { 2 } = 5048x2+69xy+25y2=50 ​

Use the discriminant to classify the graph.​ x2+xy+10y2+x+y−10=0x ^ { 2 } + x y + 10 y ^ { 2 } + x + y - 10 = 0x2+xy+10y2+x+y−10=0 ​

Consider the equation.​ 6x2−3xy+6y2−25=06 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 06x2−3xy+6y2−25=0 ​ Without calculating,explain how to rewrite the equation so that it does not have an xyx yxy -term. ​

Consider the equation.​ 10x2−5xy+10y2−81=010 x ^ { 2 } - 5 x y + 10 y ^ { 2 } - 81 = 010x2−5xy+10y2−81=0 ​ Explain how to identify the graph of the equation. ​

Identify the conic by writing the equation in standard form. 8x2+8y2+32x+144y+520=08 x ^ { 2 } + 8 y ^ { 2 } + 32 x + 144 y + 520 = 08x2+8y2+32x+144y+520=0

Use the discriminant to classify the graph.​ x2−10xy−9y2+8x−33=0x ^ { 2 } - 10 x y - 9 y ^ { 2 } + 8 x - 33 = 0x2−10xy−9y2+8x−33=0 ​

Select the graph of the following equation,showing both sets of axes.​ (y′)22−(x′)22=1\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 2 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 2 } = 12(y′)2​−2(x′)2​=1 ​

Select the graph of degenerate conic.​ 9x2−2xy+9y2=09 x ^ { 2 } - 2 x y + 9 y ^ { 2 } = 09x2−2xy+9y2=0 ​

Determine the angle θ\thetaθ through which the axes are rotated.Round your answer to two decimal places.​ 5x2−4xy+6y2=95 x ^ { 2 } - 4 x y + 6 y ^ { 2 } = 95x2−4xy+6y2=9 ​

Use the Quadratic Formula to solve for yyy .​ x2−5xy−4y2+3x−32=0x ^ { 2 } - 5 x y - 4 y ^ { 2 } + 3 x - 32 = 0x2−5xy−4y2+3x−32=0 ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Hyperbolas

Parametric Equations

Polar Coordinates

Polar Equations of Conics

Graphs of Polar Equations

Filters

Select the graph of degenerate conic. $x ^ { 2 } + 2 x y + y ^ { 2 } - 9 = 0$

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places. $48 x ^ { 2 } + 69 x y + 25 y ^ { 2 } = 50$

Use the discriminant to classify the graph. $x ^ { 2 } + x y + 10 y ^ { 2 } + x + y - 10 = 0$

Consider the equation. $6 x ^ { 2 } - 3 x y + 6 y ^ { 2 } - 25 = 0$ Without calculating,explain how to rewrite the equation so that it does not have an $x y$ -term.

Consider the equation. $10 x ^ { 2 } - 5 x y + 10 y ^ { 2 } - 81 = 0$ Explain how to identify the graph of the equation.

Identify the conic by writing the equation in standard form. $8 x ^ { 2 } + 8 y ^ { 2 } + 32 x + 144 y + 520 = 0$

Use the discriminant to classify the graph. $x ^ { 2 } - 10 x y - 9 y ^ { 2 } + 8 x - 33 = 0$

Select the graph of the following equation,showing both sets of axes. $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 2 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 2 } = 1$

Select the graph of degenerate conic. $9 x ^ { 2 } - 2 x y + 9 y ^ { 2 } = 0$

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places. $5 x ^ { 2 } - 4 x y + 6 y ^ { 2 } = 9$

Use the Quadratic Formula to solve for $y$ . $x ^ { 2 } - 5 x y - 4 y ^ { 2 } + 3 x - 32 = 0$