Analytic Geometry Quizzes: Master Coordinate Systems, Lines, and Planes

Question 1

Which of the following is the equation of the line with a slope of -2 and a y-intercept of 3?

Accepted Answer

y = -2x + 3

Answer

y = 2x + 3

Answer

y = -2x - 3

Answer

y = 2x - 3

Question 2

Which of the following is a conic section?

Accepted Answer

x^2 + y^2 = 4

Answer

x = y^2

Answer

y = 2x + 3

Answer

x + y = 5

Question 3

Identify the matrix representing a 90-degree clockwise rotation about the origin.

Accepted Answer

[[0, -1], [1, 0]]

Answer

[[0, 1], [1, 0]]

Answer

[[1, 0], [0, -1]]

Answer

[[1, 0], [0, 1]]

Question 4

Which of the following is the equation of the line with a slope of 2 and a y-intercept of -3?

Accepted Answer

y = 2x - 3

Answer

y = -2x + 3

Answer

y = x + 2

Answer

y = -x - 2

Question 5

Which of the following transformations will translate the point (2, -3) 4 units to the right and 2 units up?

Accepted Answer

T(4, 2)

Answer

T(-4, -2)

Answer

T(-2, 4)

Answer

T(2, -4)

Question 6

Find the coordinates of the point that divides the line segment connecting the points (1, 2) and (5, 8) in a ratio of 2:1.

Accepted Answer

(3, 5)

Answer

(2, 4)

Answer

(4, 6)

Answer

(5, 7)

Question 7

Determine the slope of the line perpendicular to the line with the equation y = 4x - 1.

Accepted Answer

-1/4

Answer

1/4

Answer

-4

Answer

4

Question 8

Find the slope of a line perpendicular to the line with equation y = 2x - 3.

Accepted Answer

-1/2

Answer

2

Answer

1/2

Answer

-2

Question 9

Which of the following represents the transformation of a figure across the y-axis?

Accepted Answer

(x, y) -> (-x, y)

Answer

(x, y) -> (y, x)

Answer

(x, y) -> (-x, -y)

Answer

(x, y) -> (y, -x)

Question 10

Identify the equation that represents an ellipse in its standard form:

Accepted Answer

(x - h)²/a² + (y - k)²/b² = 1

Answer

y = mx + b

Answer

x²/a² - y²/b² = 1

Answer

x² + y² = r²

Question 11

Find the angle between the vectors <1, 2, 3> and <2, 1, -1>.

Accepted Answer

30.96°

Answer

45°

Answer

90°

Answer

120°

Question 12

Find the equation of the line passing through points (2, 3) and (6, 9).

Accepted Answer

y = 2x + 1

Answer

y = x + 1

Answer

y = x/2 + 2

Answer

y = 3x + 1

Question 13

Find the equation of the sphere with center (2, 3, 4) and radius 5.

Accepted Answer

(x - 2)² + (y - 3)² + (z - 4)² = 25

Answer

(x + 2)² + (y + 3)² + (z + 4)² = 25

Answer

(x - 2)² + (y - 3)² + (z - 4)² = 5

Answer

(x + 2)² + (y + 3)² + (z + 4)² = 5

Question 14

What is the dot product of the vectors (3, 4, 5) and (1, 2, 3)?

Accepted Answer

22

Answer

11

Answer

33

Answer

44

Question 15

Find the distance between the points (-2, 5) and (4, -3).

Accepted Answer

10

Answer

5

Answer

9

Answer

12

Question 16

Find the equation of the plane that passes through the points (1, 2, 3), (2, 3, 5), and (3, 4, 7).

Accepted Answer

x + 2y + 3z = 14

Answer

x + y + z = 6

Answer

2x + 3y + 4z = 22

Answer

3x + 4y + 5z = 30

Question 17

What is the slope of a line perpendicular to the line y = 2x + 3?

Accepted Answer

-1/2

Answer

1/2

Answer

-2

Answer

2

Question 18

Which transformation is represented by the matrix [1 0 0 | 0 1 0 | 2 3 1]?

Accepted Answer

Translation 2 units in the x-direction and 3 units in the y-direction.

Answer

Rotation 90 degrees counterclockwise around the z-axis.

Answer

Reflection over the x-axis.

Answer

Scaling by a factor of 2 in the x-direction and 3 in the y-direction.

Question 19

Find the determinant of the matrix [4 2 | -1 3].

Accepted Answer

14

Answer

6

Answer

10

Answer

18

Question 20

Find the intersection of the line x = 2 and the plane y + z = 5.

Accepted Answer

(2, 3, 0)

Answer

(0, 3, 2)

Answer

(0, 2, 3)

Answer

(2, 0, 3)

Question 21

Give the parametric equation of the line through the point (1, 2) with direction vector <2, 1>.

Accepted Answer

x = 1 + 2t, y = 2 + t

Answer

x = 2t, y = 1 + t

Answer

x = 1 + t, y = 2 - t

Answer

x = 2 + t, y = 1 + 2t

Question 22

Determine the volume of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1).

Accepted Answer

1/6

Answer

1/2

Answer

1/3

Answer

1

Question 23

What is the slope of the line that is perpendicular to the line y = -2x + 5?

Accepted Answer

1/2

Answer

-1/2

Answer

2

Answer

-2

Question 24

Which of the following transformations will map the point (-3, 2) to (2, 7)?

Accepted Answer

Translation: (x + 5, y + 5)

Answer

Rotation: 90 degrees clockwise about the origin

Answer

Translation: (x - 5, y - 5)

Question 25

Which of the following equations represents a parabola opening upwards?

Accepted Answer

y = x^2 + 2x + 1

Answer

y = x^2 - 2x + 1

Answer

y = -x^2 + 2x + 1

Question 26

Identify the vertex of the parabola with the equation y = (x - 2)^2 + 3.

Accepted Answer

(2, 3)

Answer

(2, 0)

Answer

(0, 3)

Answer

(0, 0)

Question 27

What is the slope of the line that passes through the points (2, 3) and (5, 9)?

Accepted Answer

2

Answer

-1/2

Answer

1/2

Answer

-2

Question 28

Which equation represents a line perpendicular to the line y = 2x + 1?

Accepted Answer

y = -1/2x + 3

Answer

y = 2x - 5

Answer

y = 1/2x + 2

Answer

y = -2x + 1

Question 29

What is the midpoint of the line segment with endpoints (2, 5) and (8, -1)?

Accepted Answer

(5, 2)

Answer

(10, 4)

Answer

(6, 6)

Answer

(3, 3)

Question 30

A line with a slope of -3 passes through the point (1, 2). What is the y-intercept of the line?

Accepted Answer

5

Answer

2

Answer

-3

Answer

-1

Question 31

What is the equation of the circle with center (2, -1) and radius 3?

Accepted Answer

(x - 2)² + (y + 1)² = 9

Answer

(x + 2)² + (y - 1)² = 3

Answer

(x - 2)² + (y + 1)² = 3

Answer

(x + 2)² + (y - 1)² = 9

Question 32

A translation maps the point (3, 4) to the point (1, 2). What is the translation vector?

Accepted Answer

(-2, -2)

Answer

(4, 6)

Answer

(2, 2)

Answer

(-4, -6)

Question 33

What is the equation of the plane that passes through the point (1, 2, 3) and is perpendicular to the vector (2, -1, 1)?

Accepted Answer

2x - y + z = 3

Answer

x - 2y + 3z = 1

Answer

x + 2y - 3z = 1

Answer

2x + y - z = 3

Question 34

Find the slope of the line passing through the points (2, 5) and (4, 9).

Accepted Answer

2

Answer

1

Answer

-1

Answer

4

Question 35

Which of the following equations represents the line with a slope of 3 and a y-intercept of 2?

Accepted Answer

y = 3x + 2

Answer

y = 2x + 3

Answer

x = 3y + 2

Answer

y = 2x - 3

Question 36

Which of the following equations represents a circle with a radius of 3 and its center at the point (2, -1)?

Accepted Answer

(x - 2)^2 + (y + 1)^2 = 9

Answer

(x - 1)^2 + (y + 2)^2 = 3

Answer

(x + 2)^2 + (y - 1)^2 = 3

Answer

(x + 1)^2 + (y + 2)^2 = 9

Question 37

Find the distance between the points (2, 4) and (-3, 0).

Accepted Answer

5√2

Answer

10

Answer

15

Answer

5

Question 38

Which of the following is the equation of the line passing through the points (2, 3) and (4, 7)?

Accepted Answer

y = 2x + 1

Answer

y = -2x + 7

Answer

y = 3x - 1

Answer

y = x + 5

Question 39

Find the distance between the points (1, 2) and (5, 8).

Accepted Answer

10

Answer

15

Answer

5

Answer

20

Question 40

Determine the midpoint of the line segment connecting the points (-3, 5) and (1, 1).

Accepted Answer

(-1, 3)

Answer

(2, 4)

Answer

(0, 2)

Answer

(-2, 2)

Question 41

Find the slope of the line perpendicular to the line with equation y = -2x + 5.

Accepted Answer

1/2

Answer

-2

Answer

-1/2

Answer

2

Question 42

Which of the following equations represents the parabola with vertex (2, 3) and focus (4, 3)?

Accepted Answer

y = (x - 2)^2 + 3

Answer

y = (x + 2)^2 + 3

Answer

y = (x + 4)^2 + 3

Answer

y = (x - 4)^2 + 3

Question 43

Find the equation of the plane passing through the points (1, 0, 2), (3, 4, 6), and (5, 2, 10).

Accepted Answer

2x + 4y - 3z = 6

Answer

2x - 4y - 3z = -6

Answer

3x + 4y + 2z = 12

Answer

x + 2y - 3z = 4

Question 44

Identify the transformation that maps the point (3, -2) to the point (-3, 2).

Accepted Answer

Reflection across the y-axis

Answer

Rotation 90 degrees counterclockwise

Answer

Reflection across the x-axis

Answer

Translation 6 units right

Question 45

Determine the composition of the transformations: first, a translation 3 units down, then a rotation 90 degrees clockwise.

Accepted Answer

Rotation 90 degrees clockwise followed by a translation 3 units down

Answer

Reflection across the y-axis followed by a translation 3 units down

Answer

Translation 3 units down followed by a rotation 90 degrees clockwise

Question 46

Find the inverse of the transformation: (x, y) -> (2x - 1, y + 3).

Accepted Answer

(x + 1/2, y - 3)

Answer

(2x + 1, y - 3)

Answer

(2x - 1, y - 3)

Answer

(x - 1/2, y + 3)