Module 07: Conic Sections and Polar Coordinate Systems – Mechanical Engineering Online Review Center

Handouts
Exercises

LINK: Conic Sections and Polar Coordinate Systems

Topics:

Conic Sections
Circle
Parabola
Ellipse
Hyperbola
Polar coordinates
Curve tracing on polar coordinates
Space Geometry
Quadric surfaces

Find the equation of circle passing through (-5, 4) and center at (5, 4).
- a. (x – 5)² + (y – 4)²= 100
- b. x² – 15x + y² – 8y = 82
- c. x² – 10x + y² – 12y = 80
- d. (x – 5)² + (y – 4)² = 120
The shortest distance from A (4, 9) to the circle x2 + y2 + 4x + 6y = 14.
- a. 13.4
- b. 12.2
- c. 15.8
- d. 10.8
What is the coordinate of the center of the curve x2 + y2 – 3x -5y – 35 = 0?
- a. (1.5, 2.5)
- b. (-2.3, 3.3)
- c. (1, 2)
- d. (2.5, 1.5)
What is the distance between center of the circle x² + y² – 7y = 0 and x² – 6x + y² – 6y – 14 = 0?
- a. 3
- b. 4
- c. 2
- d. 5
What is the radius of a circle with the following equation? x² – 8x + y² – 6y – 15 = 0.
- a. 6.3
- b. 5.3
- c. 4.2
- d. 2.8
Find the center of the circle given by the equation x² + y² – 6x + 3y = 10.
- a. (-2, 1.5)
- b. (1.5, 2.5)
- c. (3.5, 1.5)
- d. (3, -1.5)
The equation of the circle on the x-y plane is given by x² – 3x + y² + 3y = 3. The center and the radius of the circle are _____, respectively.
- a. C(2, 2.5), r=2.2
- b. C(-1.5, -1.5), r=2.1
- c. C(1.5, -1.5), r=2.7
- d. C(2.5, -2.5), r=2.5
Find the standard equation of a circle whose center is at (2, 3) and passes through (6, 5).
- a. x² + y² – 5x – 8y + 6 = 0
- b. x² + y² + 6x – 9y + 6 = 0
- c. x² + y² – 4x – 6y – 7 = 0
- d. x² + y² – 8x – 9y + 7 = 0
What is the radius of the circle described by x² – 3x + y² – 4y – 6 = 0?
- a. 4.8
- b. 2.2
- c. 5.0
- d. 3.5
The length of the latus rectum for the ellipse x²/49 + y²/25 = 1 is equal to ____.
- a. 7.1
- b. 6.2
- c. 8.1
- d. 5.4
Find the equation of the directrix of the parabola y² = 18x
- a. 4
- b. -4.5
- c. 4.5
- d. -4
The focus of the parabola y² = 24x is at _____.
- a. (5, 0)
- b. (3, 0)
- c. (6, 0)
- d. (2, 0)
The equation x2 – 10y2 + 8x – 4y + 8 = 0 is:
- a. Hyperbola
- b. Circle
- c. Parabola
- d. Ellipse
The equation y² + y – 4x + 8 = 0 opens:
- a. to the left
- b. to the right
- c. upward
- d. downward
The equation y² + y – 4x + 8 = 0 is:
- a. Hyperbola
- b. Circle
- c. Parabola
- d. Ellipse
If B² – 4AC = 0, then the equation of conic is:
- a. Hyperbola
- b. Circle
- c. Ellipse
- d. Parabola
An ellipse has a major axis of 10 and minor axis of 8. Find the eccentricity of the ellipse.
- a. 0.6
- b. 0.5
- c. 0.7
- d. 0.8
What is the equation of an ellipse with eccentricity (35/50)^0.5 and ends of minor axis at (0, ±5).
- a. (x²/280) + (y²/28) = 1
- b. (3x²/250) + (y²/25) = 1
- c. (5x²/280) + (y²/20) = 1
- d. (9x²/200) + (y²/38) = 1
What is the equation of an ellipse with foci at (±5, 0) and major axis of length 16.
- a. 40x² – 65y² = 2652
- b. 39x² + 60y² = 2229
- c. 39x² + 64y² = 2496
- d. 35x² + 63y² = 2042
What conic section is described by the following equation 10x² + 18y² + 54x – 64y = -1?
- a. Hyperbola
- b. Circle
- c. Ellipse
- d. Parabola
Find the equation of parabola with focus at (-2, 3) and directrix at y = -3.
- a. (x-2)² = 10y
- b. (x+3)² = -8y
- c. (x-1)² = 16y
- d. (x+2)² = 12y
Find the equation of parabola with vertex at (3, 4) and directrix at x = 6.
- a. (x-4)² = 12(y+3)
- b. (y+4)² = -12(x+3)
- c. (y-4)² = -12(x-3)
- d. (x-5)² = 12(y-3)
Find the eccentricity of a parabola whose equation is y² = 8(x – 7).
- a. 0.30
- b. 0.81
- c. 1.0
- d. 0.52
Find the angle between vectors V1(4i – j + 3k) and V2(i – 3j – 5k)
- a. 122.4 deg
- b. 136.2 deg
- c. 105.4 deg
- d. 148.7 deg
Given the vector V = i + 3j + k, what is the angle between V and the x – axis?
- a. 65.35 deg
- b. 43.81 deg
- c. 57.51 deg
- d. 72.45 deg
A sphere has a radius of 5m and center at (12, 12, 12). What is the surface area.
- a. 314.2 m²
- b. 221.28 m²
- c. 368.11 m²
- d. 287.27 m²
What is the distance from the plane 3x + 3y – 2z – 7 = 0 to the point (3, 3, -5)?
- a. 3.8
- b. 4.5
- c. 2.1
- d. 5.7
Find the radius of sphere passing through point (2, 6, 3)
- a. 6
- b. 5
- c. 7
- d. 8
What is the distance between two points (2, -4, 10) and (-5, 6, -8)?
- a. 18.3
- b. 16.2
- c. 25.8
- d. 21.7
What acute angle between vectors V1 (2, 3, 2) and V2 (3, 2, 4) based at the origin?
- a. 20.8 deg
- b. 28.7 deg
- c. 25.7 deg
- d. 22.8 deg
Find the distance between points in polar coordinates (6, 30^o) and (3, 80^o)
- a. 3.8
- b. 6.3
- c. 5.8
- d. 4.7
Find the length of vector (2, 4, -5).
- a. 8.8
- b. 5.5
- c. 6.7
- d. 7.5
The equation x² + 5y + 3xy + 4x – 3y – 12 = 0 is an equation of ____.
- a. Parabola
- b. Circle
- c. Hyperbola
- d. Ellipse
Find the x and y coordinate of (8, 40^o)
- a. (6.1, 5.1)
- b. (4.3, 6.6)
- c. (7.9, 5.0)
- d. (6.1, 4.5)
Find the polar form of x² + y² – 4x + 8y = 0
- a. r = 4(cosθ – 2sinθ)
- b. r = 6(cosθ + 2sinθ)
- c. r = -3(2cosθ – sinθ)
- d. r = 4(2cosθ – cos²θ)
The angle between vector V=3i – 6j + 8k and the x, y, and z axis is _____, respectively.
- a. 40 deg
- b. 45 deg
- c. 32 deg
- d. 38 deg
Find the eccentricity of an ellipse x² + 6y² = 8.
- a. 0.21
- b. 0.99
- c. 0.76
- d. 0.38
Find the eccentricity of hyperbola 10x² – 6y² = -40.
- a. 2.8
- b. 1.3
- c. 0.68
- d. 1.8
Find the equation of parabola with focus at (0, 6) and directrix at x=8.
- a. y² + 10y – 12x – 33 = 0
- b. y² – 8y – 14x + 28 = 0
- c. 2y² – 17y – 13x – 35 = 0
- d. y² – 12y + 16x – 28 = 0

Answer Key (Handwritten)