Classifying conic sections worksheet with equations and multiple-choice answers.

To solve the problem of classifying conic sections from their equations, we need to analyze each equation and determine whether it represents a circle, ellipse, parabola, or hyperbola. The general form of a conic section is:

\[
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
\]

The type of conic section can be determined by examining the coefficients \(A\), \(B\), and \(C\) using the discriminant \(B^2 - 4AC\):

1. Circle: \(A = C\) and \(B = 0\).
2. Ellipse: \(B^2 - 4AC < 0\) and \(A \neq C\).
3. Parabola: \(B^2 - 4AC = 0\).
4. Hyperbola: \(B^2 - 4AC > 0\).

Let's classify each equation step by step.

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Question 1: \(4x^2 - 9y^2 + 54x - 101 = 0\)

- Here, \(A = 4\), \(B = 0\), and \(C = -9\).
- Calculate the discriminant:
\[
B^2 - 4AC = 0^2 - 4(4)(-9) = 0 + 144 = 144
\]
Since \(B^2 - 4AC > 0\), this is a hyperbola.
- Answer: B (Hyperbola)

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Question 2: \(3x^2 + 2y^2 - 12x + 4y - 46 = 0\)

- Here, \(A = 3\), \(B = 0\), and \(C = 2\).
- Calculate the discriminant:
\[
B^2 - 4AC = 0^2 - 4(3)(2) = 0 - 24 = -24
\]
Since \(B^2 - 4AC < 0\), this is an ellipse.
- Answer: A (Ellipse)

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Question 3: \(-x^2 - 6x + 2y - 17 = 0\)

- Here, \(A = -1\), \(B = 0\), and \(C = 0\).
- Calculate the discriminant:
\[
B^2 - 4AC = 0^2 - 4(-1)(0) = 0
\]
Since \(B^2 - 4AC = 0\), this is a parabola.
- Answer: A (Parabola)

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Question 4: \(9x^2 + 16y^2 + 36x - 108 = 0\)

- Here, \(A = 9\), \(B = 0\), and \(C = 16\).
- Calculate the discriminant:
\[
B^2 - 4AC = 0^2 - 4(9)(16) = 0 - 576 = -576
\]
Since \(B^2 - 4AC < 0\), this is an ellipse.
- Answer: C (Ellipse)

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Question 5: \(x^2 + y^2 - 2x - 4y - 13 = 0\)

- Here, \(A = 1\), \(B = 0\), and \(C = 1\).
- Calculate the discriminant:
\[
B^2 - 4AC = 0^2 - 4(1)(1) = 0 - 4 = -4
\]
Since \(B^2 - 4AC < 0\) and \(A = C\), this is a circle.
- Answer: D (Circle)

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Question 6: \(x^2 + y^2 + 2y - 15 = 0\)

- Here, \(A = 1\), \(B = 0\), and \(C = 1\).
- Calculate the discriminant:
\[
B^2 - 4AC = 0^2 - 4(1)(1) = 0 - 4 = -4
\]
Since \(B^2 - 4AC < 0\) and \(A = C\), this is a circle.
- Answer: B (Circle)

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Final Answers:

1. B (Hyperbola)
2. A (Ellipse)
3. A (Parabola)
4. C (Ellipse)
5. D (Circle)
6. B (Circle)

\[
\boxed{B, A, A, C, D, B}
\]