PRE CALCULUS Conic Sections and Parametric Equations 2 — Questions and Answers
Question 1: What is the eccentricity of a circle?
- 0 (Correct answer)
- 1
- Greater than 1
- Between 0 and 1
Correct answer: 0
A circle has eccentricity e = 0, meaning both foci coincide at the center.
Question 2: For the hyperbola x²/16 − y²/9 = 1, what are the equations of the asymptotes?
- y = ±(3/4)x (Correct answer)
- y = ±(4/3)x
- y = ±4x
- y = ±3x
Correct answer: y = ±(3/4)x
The asymptotes of x²/a² − y²/b² = 1 are y = ±(b/a)x, so y = ±(3/4)x.
Question 3: Which parametric equations trace a circle of radius 3 centered at the origin?
- x = 3cos(t), y = 3sin(t) (Correct answer)
- x = 3t, y = 3t
- x = 3 + cos(t), y = 3 + sin(t)
- x = cos(3t), y = sin(3t)
Correct answer: x = 3cos(t), y = 3sin(t)
x = 3cos(t) and y = 3sin(t) satisfy x² + y² = 9, tracing a circle of radius 3.
Question 4: The directrix of the parabola x² = 8y is:
- y = −2 (Correct answer)
- y = 2
- x = −2
- x = 2
Correct answer: y = −2
From x² = 4py with 4p = 8, p = 2; the directrix is y = −p = −2.
Question 5: To eliminate the parameter t from x = t + 1, y = t² − 3, what is the Cartesian equation?
- y = (x−1)² − 3 (Correct answer)
- y = x² − 3
- y = x − 4
- y = (x+1)² − 3
Correct answer: y = (x−1)² − 3
Solving x = t + 1 gives t = x − 1; substituting into y = t² − 3 yields y = (x−1)² − 3.
Question 6: How many foci does an ellipse have?
- 2 (Correct answer)
- 1
- 0
- 4
Correct answer: 2
Every ellipse has exactly two foci located on the major axis, equidistant from the center.
PRE CALCULUS Conic Sections and Parametric Equations 2
What is the eccentricity of a circle?