Question 1

Solve the problem.
-The path of a comet can be modeled by the equation $\frac { x ^ { 2 } } { ( 1,169.7 ) ^ { 2 } } - \frac { y ^ { 2 } } { ( 26 ) ^ { 2 } } = 1$ where x and y are measured in AU (astronomical units). Determine the distance (in AU) at perihelion. Round to one decimal place.
A) 1,170.0 AU
B) 0.3 AU
C) 0.6 AU
D) 1,169.7 AU

Accepted Answer

The answer of Solve the problem.
-The path of a comet...

Question 2

Solve the problem.
- Radio signals emitted from points (10, 0) and (-10, 0) indicate that a plane is 8 miles closer to (10,0) than to (-10, 0). Find an equation of the hyperbola that passes through the plane's location with foci (10, 0) and (-10, 0). All units are in miles.
Radio signals emitted from points (0, 8) and (0, -8) indicate that a plane is 4 miles farther from (0, 8) than (0, -8). Find an equation of the hyperbola that passes through the plane's location with foci (0,8) and (0, -8).
From the figure, the plane is located in the fourth quadrant of the coordinate system. Solve the system of equations defining the two hyperbolas for the point of intersection in the fourth quadrant. This is the location o the plane. 
Round the coordinates to the nearest tenth of a mile.

A) a. $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 84 } = 1$
b. $\frac { y ^ { 2 } } { 4 } - \frac { x ^ { 2 } } { 60 } = 1$
c. $( 4.1,2.3 )$

B) $\mathrm { a } \cdot \frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 84 } = 1$
b. $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 60 } = 1$
c. $( 4.1 , - 2.3 )$

C) a. $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 84 } = 1$
b. $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 60 } = 1$
c. $( - 2.3,4.1 )$

D) a. $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 84 } = 1$
b. $\frac { y ^ { 2 } } { 4 } - \frac { x ^ { 2 } } { 60 } = 1$
c. $( 4.1 , - 2.3 )$

Accepted Answer

The answer of Solve the problem.
- Radio signals emitted from...

Question 3

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Given $y ^ { 2 } = 4 p x$ with p < 0, the parabola opens (upward/downward/ left/right).

Accepted Answer

The answer of Write the word or phrase that best...

Question 4

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-A is the set of all points in a plane that are equidistant from a fixed line (called the________) and a fixed point (called the________ ).

Accepted Answer

The answer of Write the word or phrase that best...

Question 5

Find the standard form of the equation of the ellipse or hyperbola shown.
- 
A) $\frac { ( x + 1 ) ^ { 2 } } { 16 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
B) $\frac { ( x + 1 ) ^ { 2 } } { 16 } - \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
C) $\frac { ( x + 1 ) ^ { 2 } } { 4 } - \frac { ( y - 1 ) ^ { 2 } } { 2 } = 1$
D) $\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y + 1 ) ^ { 2 } } { 4 } = 1$

Accepted Answer

The answer of Find the standard form of the equation...

Question 6

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-If the directrix is horizontal, then the parabola opens (horizontally/vertically).

Accepted Answer

The answer of Write the word or phrase that best...

Question 7

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Given a parabola defined by $y ^ { 2 } = 4 p x$ , the directrix is given by $x = - p$ and the focus is given by the ordered pair .

Accepted Answer

The answer of Write the word or phrase that best...

Question 8

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The length of the latus rectum is called the________ diameter.

Accepted Answer

The answer of Write the word or phrase that best...

Question 9

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The standard form of an equation of a parabola with vertex (0, 0) and vertical axis of symmetry is________
.

Accepted Answer

The answer of Write the word or phrase that best...

Question 10

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Given $x ^ { 2 } = 4 p y$ with p > 0, the parabola opens (upward/downward/ left/right).

Accepted Answer

The answer of Write the word or phrase that best...

Question 11

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The line segment perpendicular to the axis of symmetry, passing through the focus and with endpoints on the parabola is called the________ ________ .

Accepted Answer

The answer of Write the word or phrase that best...

Question 12

Solve the system of equations.
-$$\begin{array} { l } 
x ^ { 2 } + 4 y ^ { 2 } = 103 \
4 x ^ { 2 } - y ^ { 2 } = 4
\end{array}$$
A) $\{ ( \sqrt { 7 } , 2 \sqrt { 6 } ) , ( - \sqrt { 7 } , 2 \sqrt { 6 } ) , ( \sqrt { 7 } , - 2 \sqrt { 6 } ) , ( - \sqrt { 7 } , - 2 \sqrt { 6 } ) \}$
B) $\{ ( 7,24 ) , ( - 7,24 ) , ( 7 , - 24 ) , ( - 7 , - 24 ) \}$
C) $\{ ( 2 \sqrt { 6 } , \sqrt { 7 } ) , ( 2 \sqrt { 6 } , - \sqrt { 7 } ) , ( - 2 \sqrt { 6 } , \sqrt { 7 } ) , ( - 2 \sqrt { 6 } , - \sqrt { 7 } ) \}$
D) $\{ ( 24,7 ) , ( - 24,7 ) , ( 24 , - 7 ) , ( - 24 , - 7 ) \}$

Accepted Answer

By solving the system of equations, we find that the solutions for $x$ and $y$ are $\pm \sqrt{7}$ and $\pm 2\sqrt{6}$, respectively.

Question 13

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The line perpendicular to the directrix and passing through the focus is called the axis of________ .

Accepted Answer

The answer of Write the word or phrase that best...

Question 14

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Given $( y - k ) ^ { 2 } = 4 p ( x - h )$ the ordered pairs representing the vertex and focus are ________and________, respectively. The directrix is the line defined by the equation________ .

Accepted Answer

The answer of Write the word or phrase that best...

Question 15

Solve the problem.
-In 1911, Ernest Rutherford discovered the nucleus of the atom. Experiments leading to this discovery involved the scattering of alpha particles by the heavy nuclei in gold foil. When alpha particles are thrust towards the
Gold nuclei, the particles are deflected and follow a hyperbolic path. Suppose the minimum distance that the alpha particles get to the gold nucleus is 6 microns, and that hyperbolic path has asymptotes of $y = \pm \frac { 1 } { 2 } x$. Determine an equation of the path of the particle shown.
 
A) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 9 } = 1$
B) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 6 } - \frac { y ^ { 2 } } { 3 } = 1$
D) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 9 } = 1$

Accepted Answer

The answer of Solve the problem.
-In 1911, Ernest Rutherford discovered...

Question 16

Find the standard form of the equation of the ellipse or hyperbola shown.
- 
A) $\frac { ( x - 2 ) ^ { 2 } } { 4 } + \frac { ( y - 1 ) ^ { 2 } } { 5 } = 1$
B) $\frac { ( x - 2 ) ^ { 2 } } { 16 } + \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$
C) $\frac { ( x + 2 ) ^ { 2 } } { 16 } + \frac { ( y + 1 ) ^ { 2 } } { 25 } = 1$
D) $\frac { ( x - 2 ) ^ { 2 } } { 16 } - \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$

Accepted Answer

The answer of Find the standard form of the equation...

Question 17

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The distance between the vertex and the focus of a parabola is called the ________length and is often represented by $$| p |$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 18

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The________ of a parabola is the point of intersection of the parabola and the axis of symmetry.

Accepted Answer

The answer of Write the word or phrase that best...

Question 19

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Given $( x - h ) ^ { 2 } = 4 p ( y - k )$ ), the ordered pairs representing the vertex and focus are________ and________, respectively. The directrix is the line defined by the equation________ .

Accepted Answer

The answer of Write the word or phrase that best...

Question 20

Solve the problem.
-Atomic particles with like charges tend to repel one another. Suppose that two beams of like-charged particles are hurled towards each other from two parallel atomic accelerators. The path
Defined by the particles is $x ^ { 2 } - 4 y ^ { 2 } = 36$ , where x and y are measured in microns. What is the
Minimum distance between the particles?
A) 4 microns
B) 12 microns
C) 36 microns
D) 6 microns

Accepted Answer

The answer of Solve the problem.
-Atomic particles with like charges...

Quiz 7: Analytic Geometry

Solve the problem. -The path of a comet can be modeled by the equation $\frac { x ^ { 2 } } { ( 1,169.7 ) ^ { 2 } } - \frac { y ^ { 2 } } { ( 26 ) ^ { 2 } } = 1$ where x and y are measured in AU (astronomical units) . Determine the distance (in AU) at perihelion. Round to one decimal place.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given $y ^ { 2 } = 4 p x$ with p < 0, the parabola opens (upward/downward/ left/right).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A is the set of all points in a plane that are equidistant from a fixed line (called the) and a fixed point (called the ).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If the directrix is horizontal, then the parabola opens (horizontally/vertically).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given a parabola defined by $y ^ { 2 } = 4 p x$ , the directrix is given by $x = - p$ and the focus is given by the ordered pair .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The length of the latus rectum is called the________ diameter.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The standard form of an equation of a parabola with vertex (0, 0) and vertical axis of symmetry is________ .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given $x ^ { 2 } = 4 p y$ with p > 0, the parabola opens (upward/downward/ left/right).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The line segment perpendicular to the axis of symmetry, passing through the focus and with endpoints on the parabola is called the .

Solve the system of equations. - $\begin{array} { l } x ^ { 2 } + 4 y ^ { 2 } = 103 \\4 x ^ { 2 } - y ^ { 2 } = 4\end{array}$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The line perpendicular to the directrix and passing through the focus is called the axis of________ .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given $( y - k ) ^ { 2 } = 4 p ( x - h )$ the ordered pairs representing the vertex and focus are and, respectively. The directrix is the line defined by the equation .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The distance between the vertex and the focus of a parabola is called the ________length and is often represented by $| p |$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The________ of a parabola is the point of intersection of the parabola and the axis of symmetry.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given $( x - h ) ^ { 2 } = 4 p ( y - k )$ ), the ordered pairs representing the vertex and focus are and, respectively. The directrix is the line defined by the equation .

Quiz 7: Analytic Geometry

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given y2=4pxy ^ { 2 } = 4 p xy2=4px with p < 0, the parabola opens (upward/downward/ left/right).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A is the set of all points in a plane that are equidistant from a fixed line (called the________) and a fixed point (called the________ ).

Find the standard form of the equation of the ellipse or hyperbola shown. -

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If the directrix is horizontal, then the parabola opens (horizontally/vertically).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given a parabola defined by y2=4pxy ^ { 2 } = 4 p xy2=4px , the directrix is given by x=−px = - px=−p and the focus is given by the ordered pair .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The length of the latus rectum is called the________ diameter.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The standard form of an equation of a parabola with vertex (0, 0) and vertical axis of symmetry is________ .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given x2=4pyx ^ { 2 } = 4 p yx2=4py with p > 0, the parabola opens (upward/downward/ left/right).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The line segment perpendicular to the axis of symmetry, passing through the focus and with endpoints on the parabola is called the________ ________ .

Solve the system of equations. - x2+4y2=1034x2−y2=4\begin{array} { l } x ^ { 2 } + 4 y ^ { 2 } = 103 \\4 x ^ { 2 } - y ^ { 2 } = 4\end{array}x2+4y2=1034x2−y2=4​

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The line perpendicular to the directrix and passing through the focus is called the axis of________ .

Find the standard form of the equation of the ellipse or hyperbola shown. -

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The distance between the vertex and the focus of a parabola is called the ________length and is often represented by ∣p∣| p |∣p∣

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The________ of a parabola is the point of intersection of the parabola and the axis of symmetry.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given $y ^ { 2 } = 4 p x$ with p < 0, the parabola opens (upward/downward/ left/right).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A is the set of all points in a plane that are equidistant from a fixed line (called the) and a fixed point (called the ).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given a parabola defined by $y ^ { 2 } = 4 p x$ , the directrix is given by $x = - p$ and the focus is given by the ordered pair .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given $x ^ { 2 } = 4 p y$ with p > 0, the parabola opens (upward/downward/ left/right).

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The line segment perpendicular to the axis of symmetry, passing through the focus and with endpoints on the parabola is called the .

Solve the system of equations. - $\begin{array} { l } x ^ { 2 } + 4 y ^ { 2 } = 103 \\4 x ^ { 2 } - y ^ { 2 } = 4\end{array}$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The distance between the vertex and the focus of a parabola is called the ________length and is often represented by $| p |$