Class 11 Mathematics Notes Chapter 11 (Conic sections) – Mathematics Book
Detailed Notes with MCQs of Chapter 11: Conic Sections. This is a crucial chapter, not just for your Class 11 exams, but also as it forms the basis for many concepts in physics (like planetary motion) and is frequently tested in various government entrance examinations. Pay close attention to the definitions, standard equations, and properties of each conic section.
Chapter 11: Conic Sections - Detailed Notes
1. Introduction
- Definition: A conic section (or simply conic) is the curve obtained by intersecting a right circular cone (specifically, a double-napped cone) with a plane.
- Types based on Intersection:
- Let β be the angle made by the intersecting plane with the axis of the cone, and α be the semi-vertical angle of the cone.
- If β = 90°, the section is a Circle.
- If α < β < 90°, the section is an Ellipse.
- If β = α, the section is a Parabola.
- If 0 ≤ β < α, the plane cuts through both nappes, and the section is a Hyperbola.
- Let β be the angle made by the intersecting plane with the axis of the cone, and α be the semi-vertical angle of the cone.
- Degenerate Conics: If the intersecting plane passes through the vertex of the cone, we can get:
- A point (when α < β ≤ 90°)
- A straight line (when α = β)
- A pair of intersecting straight lines (when 0 ≤ β < α)
2. Circle
- Definition: A circle is the set of all points in a plane that are equidistant from a fixed point (the center) in the plane. The fixed distance is called the radius.
- Standard Equation: The equation of a circle with center (h, k) and radius r is:
(x - h)² + (y - k)² = r² - Special Case: If the center is the origin (0, 0), the equation becomes:
x² + y² = r² - General Equation: The equation x² + y² + 2gx + 2fy + c = 0 represents a circle if g² + f² - c > 0.
- Center: (-g, -f)
- Radius: √(g² + f² - c)
3. Parabola
-
Definition: A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix) in the plane.
-
Key Terms:
- Axis: The line passing through the focus and perpendicular to the directrix.
- Vertex: The point where the parabola intersects its axis. It is the midpoint between the focus and the point where the axis intersects the directrix.
- Focus (S): The fixed point.
- Directrix (L): The fixed line.
- Latus Rectum: A chord passing through the focus, perpendicular to the axis. Its endpoints lie on the parabola.
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Standard Equations (Vertex at Origin (0,0)):
Equation Focus (S) Directrix (L) Axis Length of Latus Rectum Shape y² = 4ax (a>0) (a, 0) x = -a y = 0 (x-axis) 4a Opens Right y² = -4ax (a>0) (-a, 0) x = a y = 0 (x-axis) 4a Opens Left x² = 4ay (a>0) (0, a) y = -a x = 0 (y-axis) 4a Opens Upwards x² = -4ay (a>0) (0, -a) y = a x = 0 (y-axis) 4a Opens Downwards -
Eccentricity (e): For any point P on the parabola, the ratio of its distance from the focus (PS) to its perpendicular distance from the directrix (PM) is constant and equal to 1. e = PS/PM = 1.
4. Ellipse
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Definition: An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (the foci) in the plane is constant. This constant sum is equal to the length of the major axis (2a).
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Key Terms:
- Foci (S, S'): The two fixed points.
- Center (C): The midpoint of the line segment joining the foci.
- Major Axis: The line segment through the foci and terminating on the ellipse. Length = 2a.
- Minor Axis: The line segment through the center, perpendicular to the major axis, and terminating on the ellipse. Length = 2b.
- Vertices: The endpoints of the major axis.
- Eccentricity (e): A measure of how "oval" the ellipse is. e = c/a, where c is the distance from the center to a focus. For an ellipse, 0 < e < 1. (If e=0, it's a circle).
- Latus Rectum: A chord passing through a focus, perpendicular to the major axis. Length = 2b²/a.
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Relationship: a² = b² + c² (where a > b)
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Standard Equations (Center at Origin (0,0)):
Equation Foci Vertices Major Axis Minor Axis Length of Latus Rectum Eccentricity (e=c/a) Condition x²/a² + y²/b² = 1 (a>b) (±c, 0) (±a, 0) Along x-axis (Length 2a) Along y-axis (Length 2b) 2b²/a √(1 - b²/a²) a²=b²+c² x²/b² + y²/a² = 1 (a>b) (0, ±c) (0, ±a) Along y-axis (Length 2a) Along x-axis (Length 2b) 2b²/a √(1 - b²/a²) a²=b²+c²
5. Hyperbola
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Definition: A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points (the foci) in the plane is constant. This constant difference is equal to the length of the transverse axis (2a).
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Key Terms:
- Foci (S, S'): The two fixed points.
- Center (C): The midpoint of the line segment joining the foci.
- Transverse Axis: The line segment joining the vertices. Length = 2a.
- Conjugate Axis: The line segment through the center, perpendicular to the transverse axis. Length = 2b.
- Vertices: The points where the hyperbola intersects the transverse axis.
- Eccentricity (e): e = c/a, where c is the distance from the center to a focus. For a hyperbola, e > 1.
- Latus Rectum: A chord passing through a focus, perpendicular to the transverse axis. Length = 2b²/a.
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Relationship: c² = a² + b²
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Standard Equations (Center at Origin (0,0)):
Equation Foci Vertices Transverse Axis Conjugate Axis Length of Latus Rectum Eccentricity (e=c/a) Condition x²/a² - y²/b² = 1 (±c, 0) (±a, 0) Along x-axis (Length 2a) Along y-axis (Length 2b) 2b²/a √(1 + b²/a²) c²=a²+b² y²/a² - x²/b² = 1 (0, ±c) (0, ±a) Along y-axis (Length 2a) Along x-axis (Length 2b) 2b²/a √(1 + b²/a²) c²=a²+b²
Summary of Eccentricity (e):
- Circle: e = 0
- Parabola: e = 1
- Ellipse: 0 < e < 1
- Hyperbola: e > 1
Multiple Choice Questions (MCQs)
Here are 10 MCQs based on the concepts we've just discussed. These are typical of what you might encounter in government exams.
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The equation x² + y² - 4x + 6y - 12 = 0 represents a circle with center:
a) (2, 3)
b) (-2, 3)
c) (2, -3)
d) (-2, -3) -
The focus of the parabola y² = 16x is:
a) (0, 4)
b) (4, 0)
c) (0, -4)
d) (-4, 0) -
The length of the latus rectum of the ellipse 9x² + 16y² = 144 is:
a) 9/2
b) 9/4
c) 16/3
d) 8/3 -
The eccentricity of the hyperbola x²/9 - y²/16 = 1 is:
a) 4/3
b) 5/3
c) 3/5
d) 3/4 -
The equation of the directrix for the parabola x² = -8y is:
a) x = 2
b) x = -2
c) y = 2
d) y = -2 -
For an ellipse x²/a² + y²/b² = 1, the relationship between a, b, and c (distance from center to focus) is:
a) c² = a² + b²
b) b² = a² + c²
c) a² = b² + c²
d) a = b + c -
A conic section with eccentricity e = 1 is a:
a) Circle
b) Parabola
c) Ellipse
d) Hyperbola -
The vertices of the hyperbola y²/9 - x²/4 = 1 are:
a) (±2, 0)
b) (±3, 0)
c) (0, ±2)
d) (0, ±3) -
The equation of a circle with center at the origin and radius 5 is:
a) x² + y² = 5
b) x² + y² = 25
c) (x-5)² + (y-5)² = 0
d) x² - y² = 25 -
The length of the major axis of the ellipse 4x² + y² = 100 is:
a) 5
b) 10
c) 20
d) 25
Answer Key:
- c) (2, -3) (Hint: General equation x² + y² + 2gx + 2fy + c = 0, center (-g, -f))
- b) (4, 0) (Hint: Compare y² = 16x with y² = 4ax => 4a = 16 => a = 4. Focus is (a, 0))
- a) 9/2 (Hint: Divide by 144: x²/16 + y²/9 = 1. Here a²=16, b²=9. Latus Rectum = 2b²/a = 29/4 = 9/2)*
- b) 5/3 (Hint: a²=9, b²=16. c² = a² + b² = 9 + 16 = 25 => c=5. e = c/a = 5/3)
- c) y = 2 (Hint: Compare x² = -8y with x² = -4ay => 4a = 8 => a = 2. Directrix is y = a)
- c) a² = b² + c²
- b) Parabola
- d) (0, ±3) (Hint: Compare y²/9 - x²/4 = 1 with y²/a² - x²/b² = 1 => a² = 9 => a = 3. Vertices are (0, ±a))
- b) x² + y² = 25 (Hint: Standard equation x² + y² = r²)
- c) 20 (Hint: Divide by 100: x²/25 + y²/100 = 1. Compare with x²/b² + y²/a² = 1. Here a²=100 => a=10. Length of major axis = 2a = 20)
Revise these concepts thoroughly. Focus on identifying the type of conic from the equation and extracting its properties like center, foci, vertices, eccentricity, and latus rectum. Good luck with your preparation!