Analytic Geometry: Master Curves and Surfaces in Math

Question 1

An ellipse has its center at (2, -3) and major and minor axes lengths of 6 and 4, respectively. Find the coordinates of the endpoints of the major axis.

Accepted Answer

(-4, -3) and (8, -3)

Answer

(2, 1) and (2, -7)

Answer

(6, -3) and (-2, -3)

Answer

(2, -3) and (6, 0)

Question 2

Determine the coordinates of the point on the circle defined by the parametric equations x = 3cos(t) and y = 3sin(t) when t = π/3.

Accepted Answer

(3√3/2, 3/2)

Answer

(0, 3)

Answer

(-3, 0)

Answer

(0, 0)

Question 3

Find the parametric equations for a circle with a radius of 3, centered at the origin. (Hint: Use sine and cosine to express the coordinates.)

Accepted Answer

x = 3cos(t), y = 3sin(t)

Answer

x = 3t, y = 3t

Answer

x = sin(t), y = cos(t)

Answer

x = 3sin(t)cos(t), y = 3cos(t)sin(t)

Question 4

Which of the following equations represents a circle with radius 5 and center at the origin?

Accepted Answer

x² + y² = 25

Answer

(x - 5)² + (y - 5)² = 25

Answer

x² + y² = 100

Answer

x² - y² = 25

Question 5

What are the parametric equations of the circle with center at the origin and radius 3?

Accepted Answer

x = 3cos(t), y = 3sin(t)

Answer

x = 3cos(t), y = -3sin(t)

Answer

x = 3sin(t), y = 3cos(t)

Answer

x = 3sin(t), y = -3cos(t)

Question 6

Which of the following surfaces is described by the equation z = x² - y²?

Accepted Answer

A paraboloid

Answer

A cone

Answer

A plane

Answer

A sphere

Question 7

Determine the polar equation of the circle with center at (2, π/4) and radius 3.

Accepted Answer

r = 3 cos(θ - π/4)

Answer

r = 3

Answer

r = 3 sin(θ - π/4)

Answer

r = 3(cos(θ) + sin(θ))

Question 8

Which set of parametric equations represents a circle centered at (2, 3)?

Accepted Answer

x = 2 + 5 cos(t), y = 3 + 5 sin(t)

Answer

x = 2 + 5 sin(t), y = 3 + 5 cos(t)

Answer

x = 5 cos(t) + 2, y = 5 sin(t) + 3

Answer

x = 5 sin(t) + 2, y = 5 cos(t) + 3

Question 9

Which of the following equations represents the standard form of a circle?

Accepted Answer

(x - h)^2 + (y - k)^2 = r^2

Answer

x^2 + y^2 + 2gx + 2fy + c = 0

Answer

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

Answer

y = mx + b

Question 10

Which of the following equations represents a circle with radius 3?

Accepted Answer

x^2 + y^2 = 3

Answer

y = 2x + 1

Answer

x - y = 5

Answer

x^2 - y^2 = 4

Question 11

What is the polar coordinate representation of the point (2, π/4)?

Accepted Answer

(2, 45°)

Answer

(2, 90°)

Answer

(-2, 90°)

Answer

(-2, 45°)

Question 12

Which of the following represents the graph of the inequality y ≤ x^2 - 1?

Accepted Answer

All points below the parabola y = x^2 - 1

Answer

All points above the parabola y = x^2 - 1

Answer

All points to the right of the parabola y = x^2 - 1

Answer

All points to the left of the parabola y = x^2 - 1

Question 13

Which of the following is the intersection of the planes x + y - z = 2 and 2x - y + 3z = 5?

Accepted Answer

A line

Answer

A point

Answer

A plane

Answer

A sphere

Question 14

Identify the equation that represents a circle:

Accepted Answer

x^2 + y^2 = r^2

Answer

x^2 - y^2 = r^2

Answer

y = x^2

Answer

r = x^2 + y^2

Question 15

Determine the parametric equations that define the parabola y = x^2 + 1:

Accepted Answer

x = t, y = t^2 + 1

Answer

x = t^2, y = t + 1

Answer

x = t^2 + 1, y = t

Answer

x = t, y = t^2 - 1

Question 16

Convert the polar equation r = 2cos(theta) to rectangular form:

Accepted Answer

x = 2cos(theta), y = 2sin(theta)

Answer

x^2 + y^2 = 4

Answer

x = 2, y = cos(theta)

Answer

x^2 = 4cos(theta), y^2 = 4sin(theta)

Question 17

Find the area of the region bounded by the curves y = x^2 and y = 2:

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 18

Identify the equation that represents an ellipse:

Accepted Answer

x^2/9 + y^2/4 = 1

Answer

x^2 - y^2 = 1

Answer

y = x^2

Answer

r = x^2 + y^2

Question 19

Which parametric equations define the line segment from (1, 2) to (3, 4)?

Accepted Answer

x = 1 + t, y = 2 + t

Answer

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

Answer

x = 2 + t, y = 4 + t

Answer

x = 3 + t, y = 4 + t

Question 20

Which of the following is not a conic section?

Accepted Answer

Square

Answer

Circle

Answer

Parabola

Answer

Hyperbola

Question 21

Convert the rectangular coordinates (3, 4) to polar coordinates.

Accepted Answer

(5, 53.13°)

Answer

(3, 4)

Answer

(4, 36.87°)

Answer

(5, 36.87°)

Question 22

Find the intersection point of the line x = 2 + t, y = 3 - t, z = 1 + 2t and the plane 2x + y - z = 5.

Accepted Answer

(3, 2, 2)

Answer

(2, 3, 3)

Answer

(1, 4, 4)

Answer

(0, 5, 5)

Question 23

Which of the following equations represents a circle with its center at (0, 0) and a radius of 5 units?

Accepted Answer

x^2 + y^2 = 25

Answer

x^2 + y^2 = 5

Answer

x + y = 5

Answer

x^2 + 2xy + y^2 = 25

Question 24

Which of the following is the parametric equation of the line passing through the point (2, 3) and having a slope of 1/2?

Accepted Answer

x = 2 + t, y = 3 + t/2

Answer

x = 2 + t, y = 3 - t/2

Answer

x = 2 - t, y = 3 + t/2

Answer

x = 2 - t, y = 3 - t/2

Question 25

Convert the polar coordinates $left(5, rac{5pi}{6}ight)$ to rectangular coordinates.

Accepted Answer

$(-2.5, 4.33)$

Answer

$left(5, rac{5pi}{6}ight)$

Answer

$(-4.33, 2.5)$

Question 26

Determine the type of conic section represented by the equation 4x^2 + 9y^2 - 24x - 54y - 11 = 0. Specify its key characteristics.

Accepted Answer

Ellipse

Answer

Circle

Answer

Hyperbola

Question 27

Identify the equation that represents a circle:

Accepted Answer

x squared plus y squared equals r squared

Answer

y equals x squared

Answer

y equals sine of x

Answer

r equals a plus b cosine of theta

Question 28

Determine the slope of the line tangent to the curve y equals x cubed at x equals 1:

Accepted Answer

3

Answer

0

Answer

1

Answer

-3

Question 29

Which set of parametric equations represents a line?

Accepted Answer

x equals t squared, y equals t

Answer

x equals t, y equals t squared

Answer

x equals cosine of t, y equals sine of t

Answer

x equals e to the power of t, y equals t

Question 30

Find the equation of the parabola with vertex at (0, 0) and focus at (0, 4).

Accepted Answer

y^2 = 8x

Answer

x^2 = 8y

Answer

y^2 = 4x

Answer

x^2 = 4y

Question 31

Which parametric equation represents a line passing through points (1, 2) and (3, 4)?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 32

Which surface is represented by the equation x^2 + y^2 - z^2 = 1?

Accepted Answer

Hyperboloid of one sheet

Answer

Sphere

Answer

Cone

Answer

Elliptic paraboloid

Question 33

Identify the standard form of the equation for a circle:

Accepted Answer

(x - h)² + (y - k)² = r²

Answer

x² + y² + 2gx + 2fy + c = 0

Answer

x² - y² = r²

Answer

xy = c

Question 34

Which of the following represents the parametric equations of the line passing through points (2, 3) and (5, 7)?

Accepted Answer

x = 2 + 3t, y = 3 + 4t

Answer

x = 5 - 3t, y = 7 + 4t

Answer

x = 3 + 2t, y = 2 + 4t

Answer

x = 5 + 3t, y = 7 - 4t

Question 35

Convert the equation of the circle with center at (2, 0) and radius 3 to its polar form:

Accepted Answer

r = 2 + 3cos(theta)

Answer

r = 2 - 3cos(theta)

Answer

r = 3 + 2cos(theta)

Answer

r = 2 + 3sin(theta)

Question 36

In the equation 9x² + 4y² - 36x - 8y + 4 = 0, what is the x-coordinate of the ellipse's center?

Accepted Answer

-2

Answer

2

Answer

4

Answer

-4

Question 37

Identify the graph that represents the inequality x² + y² < 9:

Accepted Answer

Interior of a circle with radius 3

Answer

Exterior of a circle with radius 3

Answer

Interior of an ellipse with semi-major axis 3

Answer

Exterior of an ellipse with semi-minor axis 3

Question 38

Calculate the surface area of a sphere with radius r:

Accepted Answer

4πr²

Answer

πr²

Answer

2πr²

Answer

3πr²

Question 39

Determine the volume of the cone with radius r and height h:

Accepted Answer

πr²h/3

Answer

πr²h

Answer

2πr²h

Answer

3πr²h

Question 40

Identify the equation of the parabola that opens upwards, has its vertex at (0, 0), and its focus at (0, 1):

Accepted Answer

y = x^2/4

Answer

y = x^2

Answer

y = 2x^2

Answer

y = -x^2

Question 41

Which of the following equations represents a circle with a radius of 4 and its center at the origin?

Accepted Answer

x^2 + y^2 = 16

Answer

x - 4 = 0

Answer

y + 4 = 0

Answer

x^2 - y^2 = 16

Question 42

Find the parametric equations of the line passing through the points (1, 2) and (3, 4).

Accepted Answer

x = 1 + t, y = 2 + t

Answer

x = 3 + t, y = 4 + t

Answer

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

Answer

x = 3 + 2t, y = 4 + 2t

Question 43

Which of the following is a characteristic of an ellipse?

Accepted Answer

The sum of the distances from any point on the ellipse to the two foci is constant.

Answer

An ellipse has two vertices.

Answer

The eccentricity of an ellipse is always greater than 1.

Answer

The minor axis of an ellipse is the longest diameter.

Question 44

Find the equation of the plane passing through the point (1, 2, 3) and with normal vector [2, -1, 3].

Accepted Answer

2x - y + 3z = 8

Answer

x + y + z = 6

Answer

3x + y - 2z = 4

Answer

4x - 2y + z = 10

Question 45

Which of the following equations represents a parabola that opens upward and has its vertex at (2, 3)?

Accepted Answer

y = (x - 2)^2 + 3

Answer

y = (x + 2)^2 + 3

Answer

y = 3 - (x - 2)^2

Answer

y = 3 + (x + 2)^2

Question 46

Determine the parametric equations for the circle with center (3, -2) and radius 5.

Accepted Answer

x = 3 + 5cos(t), y = -2 + 5sin(t)

Answer

x = 5cos(t), y = -2 + 5sin(t)

Answer

x = 3cos(t), y = -2 + 5sin(t)

Answer

x = 3 + 5sin(t), y = -2 + 5cos(t)

Question 47

Find the equation of the straight line that goes through (1, -2) and (-3, 4).

Accepted Answer

y = -x + 1

Answer

y = x - 1

Answer

y = 2x + 2

Answer

y = -2x - 2

Question 48

Identify the type of conic section represented by the equation 4x^2 - 9y^2 = 36.

Accepted Answer

Hyperbola

Answer

Ellipse

Answer

Parabola

Answer

Circle

Question 49

Which of the following equations represents a parabola with its vertex at the origin and a vertical axis of symmetry?

Accepted Answer

y = x^2

Answer

y = x^2 + 2x + 1

Answer

y = x^2 - 4x + 4

Answer

y = -x^2 + 3x - 2

Question 50

Find the parametric equation of a circle with radius 5 and center at (2, -3).

Accepted Answer

x = 2 + 5cos(t), y = -3 + 5sin(t)

Answer

x = 5cos(t), y = -3 + 5sin(t)

Answer

x = 2 + 5cos(t), y = -3 + 5cos(t)

Answer

x = 5cos(t), y = 2 + 5sin(t)

Question 51

Write the equation of the ellipse with major axis on the y-axis and foci at (0, ±3).

Accepted Answer

y^2/9 + x^2/16 = 1

Answer

x^2/9 + y^2/16 = 1

Answer

x^2/16 + y^2/9 = 1

Answer

y^2/16 + x^2/9 = 1

Question 52

Write the equation of the sphere with center at (-1, 2, 3) and radius 4.

Accepted Answer

(x + 1)^2 + (y - 2)^2 + (z - 3)^2 = 4

Answer

(x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4

Answer

(x + 1)^2 + (y + 2)^2 + (z - 3)^2 = 16

Answer

(x - 1)^2 + (y + 2)^2 + (z + 3)^2 = 16

Question 53

Find the surface area of the cone generated by rotating the line segment from (0, 0) to (0, 5) about the y-axis.

Accepted Answer

10π

Answer

5π

Answer

15π

Answer

20π

Question 54

Determine the equation of the plane passing through the points (1, 0, 1), (3, 2, -1), and (-2, 1, 3).

Accepted Answer

x + 2y - 3z = 1

Answer

x + y - z = 1

Answer

x - y + z = 1

Answer

x - 2y + 3z = 1

Question 55

Determine the parametric equation of the cylinder x² + y² = 4.

Accepted Answer

x = 2cos(t), y = 2sin(t), z = 0

Answer

x = 2cos(t), y = 2sin(t), z = t

Answer

x = 2t, y = 2t, z = 0

Answer

x = 2t, y = 2t, z = t

Question 56

Find the equation of the tangent plane to the surface z = xy at the point (2, 1, 2).

Accepted Answer

x + y - 2z = 0

Answer

x - y + 2z = 0

Answer

2x + y - z = 0

Answer

2x - y + z = 0

Question 57

Which equation represents a circle with a center at the origin and a radius of 5?

Accepted Answer

x^2 + y^2 = 25

Answer

x^2 - y^2 = 25

Answer

(x - 5)^2 + y^2 = 25

Question 58

What is the equation of the parabola that opens upward and has its vertex at (0, 2) and focus at (0, 4)?

Accepted Answer

y = x^2 + 2

Answer

y = x^2 - 2

Answer

y = -x^2 + 2

Answer

y = 2x^2 + 2

Question 59

Determine the equation of the parabola with its vertex at (2, 3) and focus at (5, 3).

Accepted Answer

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

Answer

y = (x + 2)^2 + 3

Answer

y = (x - 2)^2 + 3

Answer

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

Question 60

Identify the parametric equation of the circle centered at the origin with a radius of 3.

Accepted Answer

x = 3cos(t), y = 3sin(t)

Answer

x = 3t, y = 3t

Answer

x = cos(t), y = 3sin(t)

Answer

x = 3cos(t), y = t

Question 61

Calculate the area of the ellipse with a semi-major axis length of 5 and a semi-minor axis length of 3.

Accepted Answer

15π

Answer

3π

Answer

25π

Answer

5π

Question 62

Classify the conic section represented by the equation 4x^2 + y^2 - 8x + 4y - 5 = 0.

Accepted Answer

Ellipse

Answer

Circle

Answer

Hyperbola

Answer

Parabola

Question 63

Find the equation of the parabola with vertex at (0, 3) and focus at (0, 5).

Accepted Answer

y = (x - 3)^2 + 1

Answer

y = (x - 3)^2 - 1

Answer

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

Answer

y = (x + 3)^2 + 1

Question 64

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

Accepted Answer

2

Answer

8

Answer

1

Answer

4