Conic Sections — Previous Year Questions (Class 11 Mathematics)
Conic sections include circles, parabolas, ellipses, and hyperbolas. Master their equations and properties.
TL;DR: Conic sections include circles, parabolas, ellipses, and hyperbolas. Master their equations and properties.
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Conic sections include circles, parabolas, ellipses, and hyperbolas. Master their equations and properties.
Conic Sections — Previous Year Questions with Solutions
Q (2023, 2 marks): Find the equation of the circle with center (3, 4) and radius 5.
Answer: Circle with center (h, k) and radius r has equation:
(x - h)² + (y - k)² = r²
With center (3, 4) and radius 5:
(x - 3)² + (y - 4)² = 5²
(x - 3)² + (y - 4)² = 25
Expanding: x² - 6x + 9 + y² - 8y + 16 = 25
x² + y² - 6x - 8y = 0
Final Answer: (x - 3)² + (y - 4)² = 25 or x² + y² - 6x - 8y = 0
Q (2022, 3 marks): Find the center and radius of the circle x² + y² - 4x + 6y - 3 = 0.
Answer: General form: x² + y² + 2gx + 2fy + c = 0
Compare with: x² + y² - 4x + 6y - 3 = 0
2g = -4, so g = -2
2f = 6, so f = 3
c = -3
Center: (-g, -f) = (2, -3)
Radius: r = √(g² + f² - c) = √(4 + 9 - (-3)) = √(4 + 9 + 3) = √16 = 4
Final Answer: Center = (2, -3), Radius = 4
Q (2023, 3 marks): Find the equation of the parabola with vertex at origin and focus at (0, 3).
Answer: Vertex at origin (0, 0), focus at (0, 3).
Since focus is on the y-axis, the parabola opens upward.
For a parabola with vertex at origin and focus at (0, a):
Equation: x² = 4ay
Here, a = 3
x² = 4(3)y
x² = 12y
Alternatively, equation is x² = 4py where p = a = 3.
Final Answer: x² = 12y
Q (2021, 3 marks): Find the vertices and foci of the ellipse (x²/25) + (y²/16) = 1.
Answer: Ellipse equation: (x²/a²) + (y²/b²) = 1 with a² = 25, b² = 16
a = 5, b = 4
Since a > b, the major axis is along the x-axis.
Vertices: (±a, 0) = (±5, 0) and (0, ±b) = (0, ±4)
For vertices on major axis: (5, 0) and (-5, 0)
Eccentricity: c² = a² - b² = 25 - 16 = 9, so c = 3
Foci: (±c, 0) = (±3, 0)
Final Answer: Vertices (±5, 0) and (0, ±4); Foci (±3, 0)
Q (2022, 2 marks): Write the equation of the hyperbola with center at origin, transverse axis along x-axis, a = 3, and b = 4.
Answer: Hyperbola with transverse axis along x-axis:
(x²/a²) - (y²/b²) = 1
With a = 3, b = 4:
(x²/9) - (y²/16) = 1
Final Answer: (x²/9) - (y²/16) = 1
Q (2023, 5 marks): Identify the type of conic section represented by 2x² + 3y² - 8x - 12y + 10 = 0 and find its center.
Answer: General equation of conic: ax² + 2hxy + by² + 2gx + 2fy + c = 0
2x² + 3y² - 8x - 12y + 10 = 0
a = 2, h = 0, b = 3, g = -4, f = -6, c = 10
Discriminant Δ = h² - ab = 0 - 2(3) = -6 < 0
Since Δ < 0 and a = b is not true (2 ≠ 3), it's an ellipse.
To find center, complete the square:
2(x² - 4x) + 3(y² - 4y) + 10 = 0
2(x² - 4x + 4 - 4) + 3(y² - 4y + 4 - 4) + 10 = 0
2[(x - 2)² - 4] + 3[(y - 2)² - 4] + 10 = 0
2(x - 2)² - 8 + 3(y - 2)² - 12 + 10 = 0
2(x - 2)² + 3(y - 2)² = 10
(x - 2)²/5 + (y - 2)²/(10/3) = 1
Center: (2, 2)
Final Answer: Ellipse with center (2, 2)
Frequently Asked Questions
What is eccentricity and what are its values for different conics?
Eccentricity (e) measures how much a conic deviates from being circular. Circle: e = 0. Ellipse: 0 < e < 1. Parabola: e = 1. Hyperbola: e > 1. For an ellipse, e = √(1 - b²/a²); for hyperbola, e = √(1 + b²/a²). Higher eccentricity means the conic is more 'elongated'.
What is the directrix of a parabola and how is it related to the focus?
The directrix is a fixed line. A parabola is the locus of points equidistant from the focus (point) and the directrix (line). For parabola x² = 4ay with vertex at origin and focus at (0, a), the directrix is the line y = -a. This property defines the parabola uniquely.
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