Multiple Choice Questions
Analytic Geometry and Conic Sections
Topic: Analytic Geometry and Conic Sections
Grade: 10
Question 1:
Which conic section is represented by the equation x^2 – 4x + y^2 + 6y – 9 = 0?
a) Circle
b) Parabola
c) Ellipse
d) Hyperbola
Answer: c) Ellipse
Explanation: The given equation can be rearranged into the standard form of an ellipse. Completing the square for both x and y terms will help identify the center and the lengths of the major and minor axes. For example, completing the square for the x terms gives (x – 2)^2, indicating a horizontal shift of 2 units to the right.
Question 2:
What is the eccentricity of the ellipse given by the equation 4x^2 + 9y^2 – 24x + 54y + 29 = 0?
a) 1/2
b) 1/3
c) √2/3
d) 2/3
Answer: c) √2/3
Explanation: To find the eccentricity of an ellipse, we need to compare the coefficients of x^2 and y^2 in the standard form of the equation. The equation can be rearranged to (x – 3)^2/9 + (y + 3)^2/4 = 1, which indicates a horizontal ellipse. The eccentricity can be found using the formula √(1 – b^2/a^2), where a and b are the lengths of the major and minor axes respectively.
Question 3:
Which conic section is represented by the equation 4x^2 – 9y^2 + 24x – 54y – 29 = 0?
a) Circle
b) Parabola
c) Ellipse
d) Hyperbola
Answer: d) Hyperbola
Explanation: The given equation can be rearranged into the standard form of a hyperbola. Completing the square for both x and y terms will help identify the center and the lengths of the transverse and conjugate axes. For example, completing the square for the x terms gives (x – 3)^2, indicating a horizontal shift of 3 units to the right.
Question 4:
What is the center of the parabola represented by the equation y = -2x^2 + 8x – 5?
a) (2, -3)
b) (-2, 3)
c) (-2, -3)
d) (2, 3)
Answer: (2, -3)
Explanation: The given equation represents a downward-opening parabola. To find the center, we need to complete the square for the x terms. The vertex form of the equation is y = a(x – h)^2 + k, where (h, k) is the vertex. Completing the square gives y = -2(x – 2)^2 – 3, indicating a vertex at (2, -3).
Question 5:
Which conic section is represented by the equation x^2 + 4y^2 + 6x – 16y + 9 = 0?
a) Circle
b) Parabola
c) Ellipse
d) Hyperbola
Answer: a) Circle
Explanation: The given equation can be rearranged into the standard form of a circle. Completing the square for both x and y terms will help identify the center and the radius of the circle. For example, completing the square for the x terms gives (x + 3)^2, indicating a horizontal shift of 3 units to the left.
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