Parabola Basics
Q. For the parabola defined by the equation x^2 = 16y, what is the distance from the vertex to the focus?
Solution
In the equation x^2 = 4py, we have 4p = 16, thus p = 4. The distance from the vertex to the focus is 4.
Correct Answer: B — 4
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Q. If a parabola opens to the left, which of the following is its standard form?
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A.
y^2 = -4px
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B.
x^2 = -4py
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C.
y^2 = 4px
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D.
x^2 = 4py
Solution
The standard form of a parabola that opens to the left is y^2 = -4px.
Correct Answer: A — y^2 = -4px
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Q. If the focus of a parabola is at (0, 2) and the directrix is y = -2, what is the equation of the parabola?
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A.
x^2 = 8y
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B.
x^2 = 4y
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C.
y^2 = 8x
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D.
y^2 = 4x
Solution
The distance from the focus to the directrix is 4, so the equation is x^2 = 8y.
Correct Answer: A — x^2 = 8y
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Q. The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?
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A.
x^2 = 12y
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B.
y^2 = 12x
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C.
x^2 = 6y
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D.
y^2 = 6x
Solution
The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.
Correct Answer: A — x^2 = 12y
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Q. The vertex of the parabola given by the equation y = 2x^2 - 4x + 1 is located at which point?
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A.
(1, -1)
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B.
(2, 0)
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C.
(1, 0)
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D.
(0, 1)
Solution
To find the vertex, use the formula x = -b/(2a). Here, a = 2, b = -4, so x = 1. Plugging x = 1 into the equation gives y = -1.
Correct Answer: A — (1, -1)
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Q. What is the axis of symmetry for the parabola given by the equation y = -3(x - 2)^2 + 5?
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A.
x = 2
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B.
y = 5
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C.
y = -3
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D.
x = -2
Solution
The axis of symmetry for a parabola in vertex form y = a(x - h)^2 + k is x = h. Here, h = 2.
Correct Answer: A — x = 2
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Q. What is the latus rectum of the parabola given by the equation y^2 = 12x?
Solution
The latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 12, so p = 3, and the latus rectum is 4p = 12.
Correct Answer: C — 6
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Q. What is the value of p for the parabola defined by the equation y^2 = 20x?
Solution
In the equation y^2 = 4px, we have 4p = 20, thus p = 5.
Correct Answer: A — 5
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Q. Which of the following points lies on the parabola y = x^2 - 4?
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A.
(2, 0)
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B.
(0, -4)
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C.
(1, -3)
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D.
(3, 5)
Solution
Substituting x = 1 into the equation gives y = 1^2 - 4 = -3, so the point (1, -3) lies on the parabola.
Correct Answer: C — (1, -3)
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