Class 11 Notes

Class 11 Mathematics Notes Chapter 11 (Chapter 11) – Examplar Problems (English) Book

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Philoid

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Examplar Problems (English)
Detailed Notes with MCQs of Chapter 11: Conic Sections from your NCERT Exemplar book. This chapter is crucial for various government exams as questions frequently appear from this topic. We will cover the core concepts, standard equations, properties, and problem-solving techniques. Pay close attention to the definitions and formulae.

Chapter 11: Conic Sections - Detailed Notes for Government Exams

1. Introduction

  • Definition: A conic section (or simply conic) is the locus of a point P that moves in a plane such that its distance from a fixed point F (called the focus) always bears a constant ratio e (called the eccentricity) to its perpendicular distance from a fixed straight line L (called the directrix), which does not contain the focus.

    • PF/PM = e, where PM is the perpendicular distance from P to the line L.

  • Alternative Definition (Section of a Cone): Conic sections are the curves obtained by intersecting a right circular double-napped cone with a plane.

    • Circle: Plane intersects the cone perpendicular to the axis (or parallel to the base). (Eccentricity e = 0)
    • Parabola: Plane intersects the cone parallel to a generator (slant height). (Eccentricity e = 1)
    • Ellipse: Plane intersects one nappe of the cone at an angle to the axis (not parallel to generator or perpendicular to axis). (Eccentricity 0 < e < 1)
    • Hyperbola: Plane intersects both nappes of the cone. (Eccentricity e > 1)
    • Degenerate Conics: If the plane passes through the vertex, we can get a point, a straight line, or a pair of intersecting straight lines.

  • General Equation of Second Degree: The equation Ax² + 2Hxy + By² + 2Gx + 2Fy + C = 0 represents a conic section. For your syllabus (and most government exams based on it), we usually deal with cases where H = 0 (no xy term).

    • Ax² + By² + 2Gx + 2Fy + C = 0
      • If A = B ≠ 0: Circle
      • If A = 0 or B = 0 (but not both): Parabola
      • If A ≠ B and A, B have the same sign (AB > 0): Ellipse
      • If A and B have opposite signs (AB < 0): Hyperbola

2. Circle (e = 0)

  • Definition: A circle is the set of all points in a plane that are equidistant from a fixed point (the centre). The constant distance is the radius.
  • Standard Equation: (x - h)² + (y - k)² = r²
    • Centre: (h, k)
    • Radius: r
  • Centre at Origin: x² + y² = r²
  • General Equation: x² + y² + 2gx + 2fy + c = 0
    • Centre: (-g, -f)
    • Radius: √(g² + f² - c)
    • Condition for a real circle: g² + f² - c > 0

3. Parabola (e = 1)

  • Definition: A parabola is the set of all points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix). PF = PM.
  • Standard Forms (Vertex at Origin):
    • (i) y² = 4ax (Right-handed)
      • Focus: S(a, 0)
      • Directrix: x = -a (or x + a = 0)
      • Axis: y = 0 (x-axis)
      • Vertex: V(0, 0)
      • Latus Rectum (Chord through focus ⊥ axis): Length = 4a. Ends: (a, 2a) and (a, -2a).
    • (ii) y² = -4ax (Left-handed)
      • Focus: S(-a, 0)
      • Directrix: x = a (or x - a = 0)
      • Axis: y = 0 (x-axis)
      • Vertex: V(0, 0)
      • Latus Rectum: Length = 4a. Ends: (-a, 2a) and (-a, -2a).
    • (iii) x² = 4ay (Upward)
      • Focus: S(0, a)
      • Directrix: y = -a (or y + a = 0)
      • Axis: x = 0 (y-axis)
      • Vertex: V(0, 0)
      • Latus Rectum: Length = 4a. Ends: (2a, a) and (-2a, a).
    • (iv) x² = -4ay (Downward)
      • Focus: S(0, -a)
      • Directrix: y = a (or y - a = 0)
      • Axis: x = 0 (y-axis)
      • Vertex: V(0, 0)
      • Latus Rectum: Length = 4a. Ends: (2a, -a) and (-2a, -a).
  • Parametric Coordinates: For y² = 4ax, any point can be represented as (at², 2at), where 't' is the parameter.

4. Ellipse (0 < e < 1)

  • Definition: An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (foci) is constant and greater than the distance between the foci. (PS + PS' = 2a)
  • Standard Forms (Centre at Origin):
    • (i) x²/a² + y²/b² = 1, where a > b > 0 (Horizontal Ellipse)
      • Major Axis: Along x-axis, Length = 2a
      • Minor Axis: Along y-axis, Length = 2b
      • Vertices: A(a, 0), A'(-a, 0)
      • Foci: S(ae, 0), S'(-ae, 0)
      • Eccentricity: e = √(1 - b²/a²) => b² = a²(1 - e²)
      • Directrices: x = a/e, x = -a/e
      • Latus Rectum: Length = 2b²/a. Ends: (±ae, ±b²/a)
    • (ii) x²/b² + y²/a² = 1, where a > b > 0 (Vertical Ellipse)
      • Major Axis: Along y-axis, Length = 2a
      • Minor Axis: Along x-axis, Length = 2b
      • Vertices: B(0, a), B'(0, -a)
      • Foci: S(0, ae), S'(0, -ae)
      • Eccentricity: e = √(1 - b²/a²) => b² = a²(1 - e²)
      • Directrices: y = a/e, y = -a/e
      • Latus Rectum: Length = 2b²/a. Ends: (±b²/a, ±ae)
  • Key Relationship: b² = a²(1 - e²) is fundamental. Remember 'a' is always the semi-major axis length.
  • Shifted Ellipse: Equation: (x-h)²/a² + (y-k)²/b² = 1. Centre is (h, k). All properties are relative to the centre (h,k).

5. Hyperbola (e > 1)

  • Definition: A hyperbola is the set of all points in a plane, the absolute difference of whose distances from two fixed points (foci) is constant and less than the distance between the foci. |PS - PS'| = 2a.
  • Standard Forms (Centre at Origin):
    • (i) x²/a² - y²/b² = 1 (Horizontal Hyperbola)
      • Transverse Axis: Along x-axis, Length = 2a
      • Conjugate Axis: Along y-axis, Length = 2b
      • Vertices: A(a, 0), A'(-a, 0)
      • Foci: S(ae, 0), S'(-ae, 0)
      • Eccentricity: e = √(1 + b²/a²) => b² = a²(e² - 1)
      • Directrices: x = a/e, x = -a/e
      • Latus Rectum: Length = 2b²/a. Ends: (±ae, ±b²/a)
    • (ii) y²/a² - x²/b² = 1 (Vertical Hyperbola)
      • Transverse Axis: Along y-axis, Length = 2a
      • Conjugate Axis: Along x-axis, Length = 2b
      • Vertices: B(0, a), B'(0, -a)
      • Foci: S(0, ae), S'(0, -ae)
      • Eccentricity: e = √(1 + b²/a²) => b² = a²(e² - 1)
      • Directrices: y = a/e, y = -a/e
      • Latus Rectum: Length = 2b²/a. Ends: (±b²/a, ±ae)
  • Key Relationship: b² = a²(e² - 1) is fundamental. 'a' is always the semi-transverse axis length.
  • Asymptotes: Lines that the hyperbola approaches at infinity. For x²/a² - y²/b² = 1, the equations are y = ±(b/a)x or x/a ± y/b = 0.
  • Shifted Hyperbola: Equation: (x-h)²/a² - (y-k)²/b² = 1. Centre is (h, k). All properties are relative to the centre (h,k).

Key Takeaways for Exams:

  • Be able to identify the type of conic from its equation.
  • Know the standard forms and be able to find the centre, vertices, foci, directrices, axes lengths, latus rectum length, and eccentricity for each type.
  • Be comfortable finding the equation of a conic given specific parameters (e.g., focus and directrix for parabola, foci and vertices for ellipse/hyperbola).
  • Understand the role of eccentricity in defining the shape of the conic.
  • Practice problems involving shifted conics (where the centre is not at the origin).

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed, similar to what you might encounter in government exams:

  1. The equation 9x² + 4y² = 36 represents:
    (A) A circle
    (B) A parabola
    (C) An ellipse
    (D) A hyperbola

  2. The centre and radius of the circle x² + y² - 4x + 6y - 12 = 0 are:
    (A) Centre (2, -3), Radius 5
    (B) Centre (-2, 3), Radius 5
    (C) Centre (2, -3), Radius √12
    (D) Centre (-2, 3), Radius √22

  3. The focus of the parabola y² = -12x is:
    (A) (3, 0)
    (B) (-3, 0)
    (C) (0, -3)
    (D) (0, 3)

  4. The eccentricity of the ellipse 16x² + 25y² = 400 is:
    (A) 3/5
    (B) 4/5
    (C) 5/3
    (D) 5/4

  5. The length of the latus rectum of the hyperbola 9x² - 16y² = 144 is:
    (A) 9/2
    (B) 9/4
    (C) 32/3
    (D) 16/3

  6. The equation of the parabola with focus (6, 0) and directrix x = -6 is:
    (A) x² = 24y
    (B) x² = -24y
    (C) y² = 24x
    (D) y² = -24x

  7. For which conic section is the eccentricity equal to 1?
    (A) Circle
    (B) Parabola
    (C) Ellipse
    (D) Hyperbola

  8. The vertices of the hyperbola y²/9 - x²/16 = 1 are:
    (A) (±4, 0)
    (B) (±3, 0)
    (C) (0, ±4)
    (D) (0, ±3)

  9. The equation of the ellipse whose foci are (±4, 0) and eccentricity is 1/3 is:
    (A) x²/144 + y²/128 = 1
    (B) x²/128 + y²/144 = 1
    (C) x²/16 + y²/9 = 1
    (D) x²/9 + y²/16 = 1

  10. The equation (x-1)² + (y+2)² = 0 represents:
    (A) A circle of radius 1
    (B) A point (1, -2)
    (C) An empty set (no locus)
    (D) A pair of lines

Answer Key for MCQs:

  1. (C) - Divide by 36: x²/4 + y²/9 = 1. A and B have same sign, A≠B.
  2. (A) - Centre (-g, -f) = (-(-4/2), -(6/2)) = (2, -3). Radius = √(g²+f²-c) = √((-2)² + 3² - (-12)) = √(4+9+12) = √25 = 5.
  3. (B) - Compares with y² = -4ax. 4a = 12 => a = 3. Focus is (-a, 0) = (-3, 0).
  4. (A) - Divide by 400: x²/25 + y²/16 = 1. Here a²=25, b²=16 (a>b). e = √(1 - b²/a²) = √(1 - 16/25) = √(9/25) = 3/5.
  5. (A) - Divide by 144: x²/16 - y²/9 = 1. Here a²=16, b²=9. Latus Rectum = 2b²/a = 2(9)/√16 = 18/4 = 9/2.
  6. (C) - Focus (a, 0) = (6, 0) => a=6. Directrix x = -a = -6. This is a right-handed parabola y² = 4ax = 4(6)x = 24x.
  7. (B) - By definition.
  8. (D) - Compares with y²/a² - x²/b² = 1. a²=9 => a=3. Vertices are (0, ±a) = (0, ±3).
  9. (A) - Foci (±ae, 0) = (±4, 0) => ae = 4. Eccentricity e = 1/3. So, a(1/3) = 4 => a = 12 => a² = 144. Now, b² = a²(1 - e²) = 144(1 - (1/3)²) = 144(1 - 1/9) = 144(8/9) = 16 * 8 = 128. Equation is x²/a² + y²/b² = 1 => x²/144 + y²/128 = 1.
  10. (B) - The sum of two squares is zero only if each term is zero. (x-1)²=0 => x=1. (y+2)²=0 => y=-2. The only point satisfying this is (1, -2). This is a degenerate circle (a point circle).

Study these notes thoroughly and practice problems from the Exemplar book. Good luck with your preparation!

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