Deck 9: Analytical Geometry

Question 1

Three non-colinear points are given and can be connected with line segments to form a triangle. Find the perimeter of the triangle (rounded to the nearest tenth as needed) and determine if the triangle is a right triangle.
P₁ = (3, 6)
P₂ = (0, 6)
P₃ = (7, -3)

Answer 1

perimeter: 24.3; not a right triangle

Question 2

Line segments ( ) and ( ) have the endpoints indicated. Use the slope formula to determine if the segments are parallel, perpendicular or simply intersect. A(-3, 4) B(5, -2)
C(-4, 3) D(3, 5)

A) parallel
B) perpendicular
C) intersect

Answer 2

intersect

Question 3

A theorem from elementary geometry states: If P₁ and P₂ are on opposite sides of a line segment and an equal distance from the endpoints, then the line through P₁ and P₂ is a perpendicular bisector of the segment. Verify this is true for the segment and points shown.

Answer 3

First show that the conditions of the statement are met.
Although it is evident from the graph that P₁ and P₂ are on opposite sides of the line segment , it can be proven using the equation of the line .
Find the slope of this line using the endpoints of , R (-6, 7) and S (2, -5) and the slope formula: and use the slope-point to find the equation of the line: It can then be shown that P₁(1, 3) is in the half-plane : and P₂(-8, -3) is in the half-plane : and they are therefore on opposite sides of the line segment .
Next, use the distance formula to show that P₁(1, 3) is an equal distance from the endpoints R (-6, 7) and S (2, -5):
d(P₁R) d(P₁S) and that P₂(-8, -3) is an equal distance from the endpoints R (-6, 7) and S (2, -5):
d(P₂R) d(P₂S) Now show that the line is a perpendicular bisector of .
Use the slope formula to find the slope of the line through P₁(1, 3) and P₂(-8, -3):

Since from first step, and are perpendicular.
Use the point-slope formula to find the equation of : Find the midpoint of : Show is on : Therefore is a perpendicular bisector of .

Question 4

A theorem from elementary geometry states: If the diagonals of a quadrilateral have equal length and bisect each other, then the quadrilateral is a rectangle. Verify the points A(-1, 4), B(5, 1), C(3, -3), and D(-3, 0) are the vertices of a rectangle.

Question 5

Line segments ( ) and ( ) have the endpoints indicated. Use the slope formula to determine if the segments are parallel, perpendicular or simply intersect. A(-1, 4) B(1, -6)
C(-5, 0) D(5, 2)

A) parallel
B) perpendicular
C) intersect

Question 6

Complete the square in both x and y to write the equation in standard form. 4x² + 16y² + 16x - 64y + 16 = 0

A)
B)
C)
D)

Question 7

Write the equation in standard form and identify the center and the values of a and b.
x² + 9y² = 36

Question 8

Identify the equation as that of an ellipse or a circle. 5(x - 5)² + 5(y + 7)² = 80.

A) ellipse
B) circle

Question 9

Write the equation in standard form and sketch the graph.
x² + 9y² = 36

Question 10

Graph the hyperbola. Label the vertices.

A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D)

(Gridlines are spaced one unit apart.)

Question 11

Complete the square in both x and y to write the equation in standard form. Then draw a complete graph of the relation and identify all important features.
9x² + 4y² - 18x + 16y - 11 = 0

Question 12

Show the figure drawn by connecting points A(-3, 1), B(4, -5), C(5, -1), and D(-2, 5) is a parallelogram (opposite sides parallel and equal in length).

Question 13

Of the following four points, three are an equal distance from the point P(3, 4) and one is not. Identify which three.
Q(6, 8) R(-1, 1) S

A) Q, R, S
B) Q, R, T
C) Q, S, T
D) R, S, T

Question 14

Find two additional points that are the same (non-vertical, non-horizontal) distance from the point P(-5, 2) as S .

Question 15

Sketch the graph of the ellipse.

A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D)

(Gridlines are spaced one unit apart.)

Question 16

Identify the equation as that of an ellipse or a circle. 9(x - 7)² + 4(y + 6)² = 36.

A) ellipse
B) circle

Question 17

Line segments ( ) and ( ) have the endpoints indicated. Use the slope formula to determine if the segments are parallel, perpendicular or simply intersect. A(-4, 2) B(3, -4)
C(-2, 5) D(5, -1)

A) parallel
B) perpendicular
C) intersect

Question 18

Sketch the graph of the ellipse.

Question 19

Three non-colinear points are given and can be connected with line segments to form a triangle. Find the perimeter of the triangle (rounded to the nearest tenth as needed) and determine if the triangle is a right triangle. P₁ = (-8, 2)
P₂ = (-8, -6)
P₃ = (7, 2)

A) perimeter: 33.0; right triangle
B) perimeter: 40.0; not a right triangle
C) perimeter: 40.0; right triangle
D) perimeter: 33.0; not a right triangle

Question 20

Graph the hyperbola. Label the vertices.

A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D)

(Gridlines are spaced one unit apart.)

Question 21

Use the following to answer questions :
y = 2x² - x - 3
Find the x- and y-intercepts (if they exist).

A) x-intercepts: ; y-intercept: (0, 3)
B) x-intercepts: ; y-intercept: (0, -3)
C) x-intercepts: ; y-intercept: (0, -3)
D) x-intercepts: none; y-intercept: (0, -3)

Question 22

Classify the equation as that of a circle, ellipse, hyperbola, or none of the above. 4x² + 9y² - 40x - 18y = -73

A) Circle
B) Ellipse
C) Hyperbola
D) None of these

Question 23

Sketch a complete graph of the equation, including asymptotes. Be sure to identify the center and vertices.

A)

B)

C)

D)

Question 24

Sketch a complete graph of the equation, including asymptotes. Be sure to identify the center and vertices.
(x - 1)² - 9(y - 2)² = 9

Question 25

Use the following to answer questions :
y = x² - 4x + 3
Find the vertex.

Question 26

Graph the hyperbola. Label the vertices.

Question 27

Sketch a complete graph of the equation, including asymptotes. Be sure to identify the center and vertices.

Question 28

For the graph given, state the location of the vertices, the equation of the transverse axis, the location of the center, and the equation of the conjugate axis. (Gridlines are spaced one unit apart.)

A) vertices: (1, 2), (1, 4); transverse axis: x = 3; center (1, 3); conjugate axis: y = 1
B) vertices: (1, 2), (1, 4); transverse axis: y = 3; center (1, 3); conjugate axis: x = 1
C) vertices: (1, 2), (1, 4); transverse axis: x = 1; center (1, 3); conjugate axis: y = 3
D) vertices: (1, 2), (1, 4); transverse axis: y = 1; center (1, 3); conjugate axis: x = 3

Question 29

Classify the equation as that of a circle, ellipse, hyperbola, or none of the above. 9x² + 9y² - 18x + 72y + 72 = 0

A) Circle
B) Ellipse
C) Hyperbola
D) None of these

Question 30

Classify the equation as that of a circle, ellipse, hyperbola, or none of the above. 4x² - 7y² = 28

A) Circle
B) Ellipse
C) Hyperbola
D) None of these

Question 31

Sketch a complete graph of the equation, including asymptotes. Be sure to identify the center and vertices.

A)

B)

C)

D)

Question 32

For the graph given, state
(a) the location of the vertices
(b) the equation of the transverse axis
(c) the location of the center
(d) the equation of the conjugate axis
Assume all coordinates are lattice points. (Gridlines are spaced one unit apart.)

Question 33

Sketch a complete graph of the equation, including asymptotes. Be sure to identify the center and vertices.
x² - 9y² - 6x - 36y - 36 = 0

Question 34

Sketch a complete graph of the equation, including asymptotes. Be sure to identify the center and vertices.

Question 35

Graph the hyperbola. Label the vertices.

A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D)

(Gridlines are spaced one unit apart.)

Question 36

Classify the equation as that of a circle, ellipse, hyperbola, or none of the above. 9x² = -49y² + 441

A) Circle
B) Ellipse
C) Hyperbola
D) None of these

Question 37

Sketch a complete graph of the equation, including asymptotes. Be sure to identify the center and vertices.
25y² - 4x² = 100

Question 38

Classify the equation as that of a circle, ellipse, hyperbola, or none of the above. 8y² - 4 = 5x² + 36

A) Circle
B) Ellipse
C) Hyperbola
D) None of these

Question 39

Classify the equation as that of a circle, ellipse, hyperbola, or none of the above. -2x² + -2y²= -20

A) Circle
B) Ellipse
C) Hyperbola
D) None of these

Question 40

Use the following to answer questions :
y = x² - 4x + 3
Find the x- and y-intercepts (if they exist).

Question 41

Use the following to answer questions :
x = -y² + 6y - 8
Sketch the graph using symmetry and a few additional points.

A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D)

(Gridlines are spaced one unit apart.)

Question 42

Use the following to answer questions :
y = 2x² - x - 3
Sketch the graph using symmetry and a few additional points or completing the square and shifting a parent function.

A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D)

(Gridlines are spaced one unit apart.)

Question 43

Use the following to answer questions :
x = -y² + 6y - 8
Find the x- and y-intercepts, if they exist.

A) x-intercepts: (2, 0), (4, 0); y-intercept: (0, -8)
B) x-intercept: (2, 0); y-intercepts: (0, -8), (0, 4)
C) x-intercept: (-8, 0); y-intercepts: (0, 2), (0, 4)
D) x-intercepts: none; y-intercept (0, -8)

Question 44

(a) Sketch x = y² + 4y + 3 using symmetry and shifts of a basic function.
(b) Find the x- and y-intercepts (if they exist).
(c) Find the vertex.
(d) State the domain and range of the relation.

Question 45

(a) Sketch x = y² - 4y using symmetry and shifts of a basic function.
(b) Find the x- and y-intercepts (if they exist).
(c) Find the vertex.
(d) State the domain and range of the relation.

Question 46

Use the following to answer questions :
x² = 28y
Find the directrix.

Question 47

Use the following to answer questions :
y² = 36x
Find the directrix.

A) y = -8
B) y = 8
C) x = -8
D) x = 8

Question 48

Use the following to answer questions :
x = y² - 2y - 3
Find the vertex.

Question 49

Use the following to answer questions :
x = y² - 2y - 3
Find the x- and y-intercepts, if they exist.

Question 50

Sketch x = (y - 1)² - 4 using symmetry and shifts of a basic function.

A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D)

(Gridlines are spaced one unit apart.)

Question 51

Use the following to answer questions :
x² = 28y
Find the vertex.

Question 52

Use the following to answer questions :
x = -y² + 6y - 8

-State the domain and range.

A) x $\in$ (- $\infty$ , 1]; y $\in$ (- $\infty$ , $\infty$ )
B) x $\in$ (- $\infty$ , $\infty$ ); y $\in$ (- $\infty$ , 3]
C) x $\in$ [3, $\infty$ ); y $\in$ (- $\infty$ , $\infty$ )
D) x $\in$ (- $\infty$ , $\infty$ ); y $\in$ (- $\infty$ , $\infty$ )

Question 53

Find the vertex, focus, and directrix for the parabola given, then use this information to sketch a complete graph (illustrate these features).
y² = -8x

Question 54

Use the following to answer questions :
y = 2x² - x - 3
Find the vertex.

A)
B)
C)
D)

Question 55

Use the following to answer questions :
x² = 28y
Find the focus.

Question 56

Use the following to answer questions :
y² = 36x
Find the focus.

A) (0, -6)
B) (0, -24)
C) (-6, 0)
D) (-24, 0)

Question 57

Use the following to answer questions :
y = 2x² - x - 3

-State the domain and range.

A) x $\in$ (- $\infty$ , $\infty$ ); y $\in$
B) x $\in$ ; y $\in$ (- $\infty$ , $\infty$ )
C) x $\in$ (- $\infty$ , $\infty$ ); y $\in$
D) x $\in$ (- $\infty$ , $\infty$ ); y $\in$ (- $\infty$ , $\infty$ )

Question 58

Use the following to answer questions :
x = -y² + 6y - 8
Find the vertex.

A) (-80, -6)
B) (-8, 6)
C) (-35, -3)
D) (1, 3)

Question 59

Use the following to answer questions :
y² = 36x
Find the vertex.

A) (0, 0)
B) (0, -5)
C) (0, -20)
D) (-20, 0)

Question 60

Identify each equation in the system as that of a line, parabola, circle, ellipse, or hyperbola, and solve the system by graphing.

Question 61

Solve using elimination.

A) (4, 0), (5, -3), (-5, -3)
B) (-4, 0), (5, -3), (5, -3)
C) (-4, 0), (5, -3), (5, 3)
D) no solutions

Question 62

List three alternate ways the point can be expressed in polar coordinates using r > 0, r < 0, and $\theta$ $\in$ [-2 $\pi$ , 2 $\pi$ ).

Question 63

Plot the point .

Question 64

Solve using the method of your choice.

A) (4, 5²⁴)
B) (-4, 5⁸)
C) (4, 5²⁴), (-4, 5⁸)
D) (4, 5²⁴), (-4, -5⁸)

Question 65

Solve the system of inequalities.

Question 66

Convert from rectangular coordinates to polar coordinates. A diagram may help.
(8, -8)

Question 67

Plot the point .

A)

B)

C)

D)

Question 68

Identify each equation in the system as that of a line, parabola, circle, ellipse, or hyperbola, and solve the system by graphing.

Question 69

Express the point shown using polar coordinates with $\theta$ in radians, 0 $\le$$\theta$ < 2? and r > 0.

Question 70

Solve using elimination.

Question 71

Solve using substitution. [Hint: Substitute for y² (not x).]

Question 72

Convert from rectangular coordinates to polar coordinates. A diagram may help.

A)
B)
C)
D)

Question 73

Plot the point .

Question 74

Solve using substitution.

A) (0, 5), (5, 0)
B) (0, 5), (-5, 0)
C) (0, -5), (5, 0)
D) (0, -5), (-5, 0)

Question 75

Solve the system of inequalities.

A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D) Ø

Question 76

Solve the system of inequalities.

Question 77

Solve the system of inequalities.

Question 78

Solve using the method of your choice.

Question 79

Solve the system of inequalities.

Question 80

Solve the system of inequalities.

A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D) Ø