Precalculus Practice Test: Conic Sections
- Conic Sections
- 10 questions
- 15 minutes
These 10 questions cover the standard equations of circles, parabolas, ellipses and hyperbolas and how to read their key features. For an ellipse, c2 = a2 − b2; for a hyperbola, c2 = a2 + b2.
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Question 1
What are the center and radius of the circle (x − 2)2 + (y + 5)2 = 49?
Correct answer: B
In (x − h)2 + (y − k)2 = r2 the center is (h, k) = (2, −5), and r2 = 49 gives r = 7. Reading 49 as the radius forgets that the equation uses r2.
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Question 2
What are the center and radius of the circle x2 + y2 − 6x + 4y − 3 = 0?
Correct answer: D
Completing the square gives (x − 3)2 + (y + 2)2 = 3 + 9 + 4 = 16, so the radius is 4. The radius √3 forgets to add the 9 and 4 used to complete the squares.
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Question 3
What is the focus of the parabola y2 = 12x?
Correct answer: A
The form y2 = 4px opens right with focus (p, 0); here 4p = 12, so p = 3. The point (0, 3) would belong to x2 = 12y, which opens up.
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Question 4
What is the directrix of the parabola x2 = −8y?
Correct answer: C
Here 4p = −8, so p = −2: the parabola opens down, the focus is (0, −2), and the directrix is on the other side of the vertex at y = 2. The line y = −2 passes through the focus.
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Question 5
What are the foci of the ellipse x225 + y29 = 1?
Correct answer: B
For an ellipse c2 = a2 − b2 = 25 − 9 = 16, so c = 4, and the foci lie on the longer, horizontal axis. The points (±5, 0) are the vertices, and √34 adds instead of subtracts.
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Question 6
What is the length of the major axis of the ellipse x216 + y236 = 1?
Correct answer: D
The larger denominator is 36 = 62, so a = 6 and the major axis is vertical with length 2a = 12. The value 6 is only the semi-major axis, and 8 is the minor axis.
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Question 7
What are the asymptotes of the hyperbola x29 − y216 = 1?
Correct answer: A
For x2a2 − y2b2 = 1 the asymptotes are y = ±(b/a)x, with a = 3 and b = 4. The slopes ±3/4 flip the ratio, and 16/9 forgets the square roots.
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Question 8
Which conic section is the graph of 4x2 + 9y2 − 36 = 0?
Correct answer: C
Dividing by 36 gives x2/9 + y2/4 = 1: both squared terms are positive with different denominators, so it is an ellipse. A circle would need equal coefficients on x2 and y2.
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Question 9
What are the foci of the hyperbola y216 − x29 = 1?
Correct answer: B
For a hyperbola c2 = a2 + b2 = 16 + 9 = 25, so c = 5, and the positive y2 term puts the foci on the y-axis. The value √7 uses the ellipse formula, and (0, ±4) are the vertices.
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Question 10
Which equation describes the circle with center (−1, 4) that passes through the point (2, 0)?
Correct answer: A
The radius is the distance 32 + (−4)2 = 5, so r2 = 25 and the center (−1, 4) gives (x + 1)2 + (y − 4)2. The equation ending in = 5 uses r instead of r2.
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