Deck 63: Rotation of Conics

Question 1

Select the graph of the following equation.​ $$y = x \pm 5$$ ​

A)​
B)​
C)​ ​
D)​
E)​

Answer 1

​

Question 2

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

A)​
B)​
C)​
D)​
E)​

Answer 2

​

Question 3

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.​ $\theta = 30 ^ { \circ }$ , $( 2,6 )$ ​

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) $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 3 } { 2 } \right)$

Answer 3

$( \sqrt { 3 } + 3,3 \sqrt { 3 } - 1 )$

Question 4

Select the graph of the following equation,showing both sets of axes.​ $$\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 8 } - \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 8 } = 1$$ ​

A)​
B)​
C)​
D)​
E)​

Question 5

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.​ $\theta = 90 ^ { \circ }$ , $( 2,2 )$ ​

A) $$( - 2,0 )$$
B) $( 0,3 )$
C) $( 2 , - 2 )$
D) $( 2,0 )$
E) $( - 2,2 )$

Question 6

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.​ $\theta = 45 ^ { \circ }$ , $( 10,10 )$ ​

A) $( - 5 \sqrt { 2 } , 0 )$
B) $( 5 \sqrt { 2 } , 0 )$
C) $( - 10 \sqrt { 2 } , 0 )$
D) $( 6 \sqrt { 2 } , 0 )$
E) $( 10 \sqrt { 2 } , 0 )$

Question 7

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $24 x ^ { 2 } + 66 x y + 13 y ^ { 2 } = 32$ ​

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

Question 8

Use a graphing utility to graph the conic.Determine the angle $$\theta$$ through which the axes are rotated.​ $x ^ { 2 } + 2 x y + y ^ { 2 } = 25$ ​

A) $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$
B) $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$
C) $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$
D) $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$
E) $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$

Question 9

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.​ $\theta = 90 ^ { \circ }$ , $( 0,3 )$ ​

A) $( 0,4 )$
B) $( - 3,3 )$
C) $( - 3,0 )$
D) $( 3,0 )$
E) $( 0,3 )$

Question 10

Use a graphing utility to graph the conic.​ $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 8$$ ​

A)​
B)​
C)​
D)​
E)​ ​

Question 11

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $29 x ^ { 2 } + 92 x y + 6 y ^ { 2 } = 51$ ​

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

Question 12

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.​ $\theta = 30 ^ { \circ }$ , $( 1,3 )$ ​

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

Question 13

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

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 14

Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form.​ $$x y - 6 = 0$$ ​

A) $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } - \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } = 1$
B) $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } = 1$
C) $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 12 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 12 } = 1$
D) $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 12 } - \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 12 } = 1$
E) $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } - \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 12 } = 1$

Question 15

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $5 x ^ { 2 } - 10 x y + 12 y ^ { 2 } + ( 5 \sqrt { 11 } - 10 ) x - ( 7 \sqrt { 11 } + 11 ) y = 93$ ​

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

Question 16

​Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places. $3 x ^ { 2 } - 9 x y + 11 y ^ { 2 } + ( 3 \sqrt { 10 } - 9 ) x - ( 5 \sqrt { 10 } + 10 ) y = 91$ ​​

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

Question 17

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

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 18

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.​ $\theta = 45 ^ { \circ }$ , $( 9,2 )$ ​

A) $\left( \frac { 11 \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } \right)$
B) $\left( - \frac { 11 \sqrt { 2 } } { 2 } , - \frac { 7 \sqrt { 2 } } { 2 } \right)$
C) $\left( \frac { 11 } { 2 } , - \frac { 7 \sqrt { 2 } } { 2 } \right)$
D) $\left( \frac { 11 \sqrt { 2 } } { 2 } , \frac { 7 \sqrt { 2 } } { 2 } \right)$
E) $\left( \frac { 11 \sqrt { 2 } } { 2 } , - \frac { 7 \sqrt { 2 } } { 2 } \right)$

Question 19

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $17 x ^ { 2 } + 72 x y - 3 y ^ { 2 } = 77$ ​

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

Question 20

Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form.​ $x y + 3 = 0$ ​

A) $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 3 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 3 } = 1$
B) $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 3 } = 1$
C) $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } = 1$
D) $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 3 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } = 1$
E) $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } + \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } = 1$

Question 21

Select the graph of degenerate conic.​ $$x ^ { 2 } + y ^ { 2 } - 2 x + 16 y + 65 = 0$$ ​

A)​
B)​
C)​
D)​
E)​

Question 22

Select the graph of the following equation.​ $x y + 4 = 0$ ​

A)​
B)​
C)​
D)​ ​
E)​

Question 23

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

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 24

Use the discriminant to classify the graph.​ $x ^ { 2 } - 10 x y - 8 y ^ { 2 } - 18 = 0$ ​

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

Question 25

Use the discriminant to classify the graph.​ $100 x ^ { 2 } - 100 x y + 25 y ^ { 2 } + 12 y = 0$ ​

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

Question 26

Use the discriminant to classify the graph.​ $16 x ^ { 2 } - 5 x y + 10 y ^ { 2 } - 44 = 0$ ​

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

Question 27

Select the graph of degenerate conic.​ $$36 x ^ { 2 } - 72 x y + 36 y ^ { 2 } = 0$$ ​

A)​
B)​
C)​
D)​
E)​

Question 28

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

A) $y = \frac { ( x + 1 ) \pm \sqrt { ( x - 1 ) ^ { 2 } - 8 \left( x ^ { 2 } - x + 4 \right) } } { 4 }$
B) $y = \frac { - ( x + 1 ) \pm \sqrt { ( x + 1 ) ^ { 2 } - 8 \left( x ^ { 2 } + x + 4 \right) } } { 4 }$
C) $y = \frac { - ( x + 1 ) \pm \sqrt { ( x + 1 ) ^ { 2 } - 8 \left( x ^ { 2 } + x - 4 \right) } } { 4 }$
D) $y = \frac { ( x + 1 ) \pm \sqrt { ( x + 1 ) ^ { 2 } - 8 \left( x ^ { 2 } + x - 4 \right) } } { 4 }$
E) $y = \frac { - ( x + 1 ) \pm \sqrt { ( x + 1 ) ^ { 2 } + 8 \left( x ^ { 2 } + x - 4 \right) } } { 4 }$

Question 29

Use the Quadratic Formula to solve for $y$ .​ $81 x ^ { 2 } - 90 x y + 25 y ^ { 2 } + 10 y = 0$ ​

A) $y = \frac { 90 x - 10 \pm \sqrt { ( 90 x - 10 ) ^ { 2 } + 8100 x ^ { 2 } } } { 50 }$
B) $y = \frac { 90 x + 10 \pm \sqrt { ( 90 x + 10 ) ^ { 2 } + 8100 x ^ { 2 } } } { 50 }$
C) $y = \frac { 90 x + 10 \pm \sqrt { ( 90 x - 10 ) ^ { 2 } - 8100 x ^ { 2 } } } { 50 }$
D) $y = \frac { 90 x - 10 \pm \sqrt { ( 90 x - 10 ) ^ { 2 } - 8100 x ^ { 2 } } } { 50 }$
E) $y = \frac { 90 x - 10 \pm \sqrt { ( 90 x + 10 ) ^ { 2 } - 8100 x ^ { 2 } } } { 50 }$

Question 30

Use the Quadratic Formula to solve for $y$ .​ $5 x ^ { 2 } + 10 x y + 11 y ^ { 2 } + 9 x - 10 y - 42 = 0$ ​

A) $y = \frac { - ( 10 x - 10 ) \pm \sqrt { ( 10 x - 10 ) ^ { 2 } - 44 \left( 5 x ^ { 2 } - 9 x - 42 \right) } } { 22 }$
B) $y = \frac { ( 10 x - 10 ) \pm \sqrt { ( 10 x - 10 ) ^ { 2 } - 44 \left( 5 x ^ { 2 } + 9 x - 42 \right) } } { 22 }$
C) $y = \frac { - ( 10 x - 10 ) \pm \sqrt { ( 10 x - 10 ) ^ { 2 } - 44 \left( 5 x ^ { 2 } + 9 x - 42 \right) } } { 22 }$ .
D) $y = \frac { - ( 10 x - 10 ) \pm \sqrt { ( 10 x - 10 ) ^ { 2 } - 44 \left( 5 x ^ { 2 } + 9 x - 42 \right) } } { 22 }$
E) $y = \frac { - ( 10 x - 10 ) \pm \sqrt { ( 10 x - 10 ) ^ { 2 } + 44 \left( 5 x ^ { 2 } + 9 x - 42 \right) } } { 22 }$

Question 31

Select the graph of degenerate conic.​ $y ^ { 2 } - 49 x ^ { 2 } = 0$ ​

A)​
B)​ ​
​
​
​
C)​
D)​
E)​

Question 32

Select the graph of the following equation​ $36 x ^ { 2 } + 72 x y + 36 y ^ { 2 } = 0$ ​

A)​
B)​
C)​
D)​
E)​

Question 33

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

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

Question 34

Use the Quadratic Formula to solve for $y$ .​ $x ^ { 2 } - 12 x y - 10 y ^ { 2 } - 22 = 0$ ​

A) $$y = \frac { 12 x \pm \sqrt { 144 x ^ { 2 } + 40 \left( x ^ { 2 } + 22 \right) } } { - 20 }$$
B) $y = \frac { 12 x \pm \sqrt { 144 x ^ { 2 } + 40 \left( x ^ { 2 } - 22 \right) } } { - 20 }$
C) $y = \frac { 12 x \pm \sqrt { 144 x ^ { 2 } + 40 \left( x ^ { 2 } - 22 \right) } } { 20 }$
D) $$y = \frac { 12 x \pm \sqrt { 144 x + 40 \left( x ^ { 2 } - 22 \right) } } { - 20 }$$
E) $y = \frac { 12 x \pm \sqrt { 144 x ^ { 2 } + 40 \left( x ^ { 2 } + 22 \right) } } { 20 }$

Question 35

Use the Quadratic Formula to solve for $$y$$ .​ $14 x ^ { 2 } - 10 x y + 9 y ^ { 2 } - 41 = 0$ ​

A) $$y = \frac { 14 x \pm \sqrt { 100 x ^ { 2 } + 36 \left( x ^ { 2 } - 9 \right) } } { - 18 }$$
B) $y = \frac { 14 x \pm \sqrt { 100 x ^ { 2 } + 36 \left( x ^ { 2 } + 9 \right) } } { 18 }$
C) $y = \frac { 10 x \pm \sqrt { 100 x ^ { 2 } + 36 \left( 14 x ^ { 2 } - 41 \right) } } { - 18 }$
D) $y = \frac { 10 x \pm \sqrt { 100 x ^ { 2 } + 36 \left( 14 x ^ { 2 } + 41 \right) } } { 18 }$
E) $y = \frac { 10 x \pm \sqrt { 100 x ^ { 2 } - 36 \left( 14 x ^ { 2 } - 41 \right) } } { 18 }$

Question 36

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

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 37

Use the Quadratic Formula to solve for $y$ .​ $x ^ { 2 } + 12 x y + 36 y ^ { 2 } - 5 x - y - 3 = 0$ ​

A) $y = \frac { - ( 36 x + 1 ) \pm \sqrt { ( 36 x - 1 ) ^ { 2 } - 144 \left( x ^ { 2 } - 5 x - 3 \right) } } { 72 }$
B) $y = \frac { - ( 12 x + 1 ) \pm \sqrt { ( 36 x + 1 ) ^ { 2 } - 144 \left( x ^ { 2 } + 5 x + 3 \right) } } { 72 }$
C) $y = \frac { - ( 36 x - 1 ) \pm \sqrt { ( 36 x - 1 ) ^ { 2 } - 144 \left( x ^ { 2 } - 5 x + 3 \right) } } { 72 }$
D) $y = \frac { ( 36 x - 1 ) \pm \sqrt { ( 36 x - 1 ) ^ { 2 } - 144 \left( x ^ { 2 } - 5 x - 3 \right) } } { 72 }$
E) $y = \frac { - ( 12 x - 1 ) \pm \sqrt { ( 12 x - 1 ) ^ { 2 } - 144 \left( x ^ { 2 } - 5 x - 3 \right) } } { 72 }$

Question 38

Use the discriminant to classify the graph.​ $$36 x ^ { 2 } - 144 x y + y ^ { 2 } - 10 x + 5 y = 0$$ ​

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

Question 39

Use the discriminant to classify the graph.​ $$5 x ^ { 2 } + 10 x y + 11 y ^ { 2 } + 9 x - 10 y - 72 = 0$$ ​

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

Question 40

Use the discriminant to classify the graph.​ $x ^ { 2 } + 6 x y + 9 y ^ { 2 } - 5 x - y - 3 = 0$ ​

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

Question 41

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

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 42

Find the lengths of the major and minor axes of the ellipse graphed by following equation.​ $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 1 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 36 } = 1$ ​

A)Length of major axis is $2 a = 2 \cdot ( - 1 ) = - 2$ Length of minor axis is $2 b = 2 \cdot ( - 6 ) = - 12$
B)Length of major axis is $2 a = 2 \cdot 1 = 2$ Length of minor axis is $2 b=2 \cdot 6=12$
C)Length of major axis is $2 a = 2 \cdot ( - 6 ) = - 12$ Length of minor axis is $2 b = 2 \cdot ( - 1 ) = - 2$
D)Length of major axis is $2 a = 2 \cdot 6 = 12$ Length of minor axis is $2 b = 2 \cdot 1 = 2$
E)None of the above

Question 43

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.
​

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 44

Find any points of intersection of the graphs algebraically.​ $\begin{array} { l } - x ^ { 2 } + y ^ { 2 } + 2 x - 7 y + 6 = 0 \\x ^ { 2 } + y ^ { 2 } - 2 x - 7 y + 6 = 0\end{array}$ ​

A)The points of intersection are $$( 2,2 )$$ and $( 3,3 )$ .
B)The points of intersection are $( - 2 , - 2 )$ and $$( 2,3 )$$ .
C)The points of intersection are $( 3,3 )$ and $( 3,3 )$ .
D)The points of intersection are $( 2,2 )$ and $( 2,3 )$ .
E)The points of intersection are $( 3,3 )$ and $( - 3 , - 3 )$ .

Question 45

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

A)​
B)​
C)​
D)​
E)​

Question 46

Identify the conic by writing the equation in standard form. $10 y ^ { 2 } - 20 x ^ { 2 } + 80 y + 360 x - 1485 = 0$

A) $$\frac { ( y - 4 ) ^ { 2 } } { \frac { 5 } { 2 } } - \frac { ( x - 9 ) ^ { 2 } } { \frac { 5 } { 4 } } = 1$$ ;hyperbola
B) $\frac { ( y + 4 ) ^ { 2 } } { \frac { 5 } { 2 } } - \frac { ( x - 9 ) ^ { 2 } } { \frac { 5 } { 4 } } = 1$ ;hyperbola
C) $\frac { ( y + 4 ) ^ { 2 } } { \frac { 97 } { 2 } } - \frac { ( x - 9 ) ^ { 2 } } { \frac { 97 } { 4 } } = 1$ ;hyperbola
D) $\frac { ( y + 4 ) ^ { 2 } } { \frac { 149 } { 5 } } + \frac { ( x - 9 ) ^ { 2 } } { \frac { 149 } { 10 } } = 1$ ;ellipse
E) $$\frac { ( y - 4 ) ^ { 2 } } { \frac { 149 } { 5 } } + \frac { ( x - 9 ) ^ { 2 } } { \frac { 149 } { 10 } } = 1$$ ;ellipse

Question 47

Consider the equation.​ $10 x ^ { 2 } - 5 x y + 10 y ^ { 2 } - 81 = 0$ ​ Explain how 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 48

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

A)​
B)​
C)​
D)​
E)​

Question 49

Identify the conic by writing the equation in standard form.​ $25 x ^ { 2 } + 4 y ^ { 2 } - 400 x + 72 y + 1824 = 0$ ​

A)​ $\frac { ( x + 4 ) ^ { 2 } } { \frac { 4 } { 8 } } + \frac { ( y - 8 ) ^ { 2 } } { \frac { 8 } { 4 } } = 1$ ;ellipse
B)​ $$\frac { ( x - 8 ) ^ { 2 } } { \frac { 8 } { 4 } } + \frac { ( y + 4 ) ^ { 2 } } { \frac { 8 } { 4 } } = 1$$ ;ellipse
C)​ $$\frac { ( x - 8 ) ^ { 2 } } { \frac { 4 } { 8 } } + \frac { ( y + 4 ) ^ { 2 } } { \frac { 8 } { 4 } } = 1$$ ;ellipse
D)​ $\frac { ( x - 8 ) ^ { 2 } } { 4 } + \frac { ( y + 9 ) ^ { 2 } } { 25 } = 1$ ;ellipse
E)​ $( 5 x - 8 ) ^ { 2 } + ( 2 y + 9 ) ^ { 2 } = - 1783$ ;circle

Question 50

Select the graph of degenerate conic.​ $$x ^ { 2 } - 14 x y + y ^ { 2 } = 0$$ ​

A)​
B)​
C)​
D)​
E)​

Question 51

Determine whether the statement is true or false? Justify your answer. ​
The graph of the equation $x ^ { 2 } + x y + k y ^ { 2 } + 12 x + 16 = 0$ where, $k$ is any constant less than $\frac { 1 } { 4 }$ ,is a hyperbola.
​

A)False.For the graph to be a hyperbola,the discriminant must be less than zero.
B)True.For the graph to be a hyperbola,the discriminant must be greater than zero.

Question 52

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $x ^ { 2 } + 2 x y + y ^ { 2 } = 10$ ​

A) $\theta = 45 ^ { \circ }$
B) $\theta = 60 ^ { \circ }$
C) $\theta = 90 ^ { \circ }$
D) $\theta = 145 ^ { \circ }$
E) $\theta = 30 ^ { \circ }$