Conic Sections Class 11 Maths Chapter Notes

What are Conic Sections?

A conic section is a curve obtained by intersecting a right circular double-napped cone with a plane. The type of curve formed depends on the angle at which the plane intersects the cone relative to its axis and generator. When the plane does not pass through the vertex of the cone, the curves formed are called non-degenerate conic sections. These include the circle, ellipse, parabola, and hyperbola. If the plane passes through the vertex, the resulting shapes are degenerate conics: a point, a straight line, or a pair of intersecting lines.

The eccentricity (denoted by 'e') is a crucial parameter that defines the type of conic section. It is the ratio of the distance from any point on the conic to the focus, and the perpendicular distance from that point to the directrix. Mathematically, it's defined as e = PF / PM, where P is a point on the conic, F is the focus, and PM is the perpendicular distance to the directrix. This value determines the shape:

• Circle: e = 0
• Ellipse: 0 < e < 1
• Parabola: e = 1
• Hyperbola: e > 1

The general equation of a conic section is a second-degree equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0. By analyzing the coefficients and the discriminant B² - 4AC, one can identify the type of conic, though this is usually covered in more advanced contexts. For CBSE Class 11, focus on the standard forms and their properties.

Key Terms and Definitions

Conic Section

A curve formed by the intersection of a plane with a double-napped right circular cone.

Eccentricity (e)

The ratio of the distance from any point on the conic to the focus, to its perpendicular distance from the directrix. It defines the shape of the conic.

Focus (F)

A fixed point used in the definition of a conic section. Each conic has one or two foci.

Directrix

A fixed line used in the definition of a conic section. Each conic has one or two directrices.

Vertex

The point(s) where the conic section intersects its axis of symmetry.

Latus Rectum

A line segment passing through a focus, perpendicular to the axis of the conic, and whose endpoints lie on the conic.

Axis of Symmetry

A line about which the conic section is symmetrical.

The Parabola (e = 1)

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). Its eccentricity e = 1. Parabolas are fundamental in physics (projectile motion, parabolic mirrors).

Standard Equations of Parabola:

1. y² = 4ax (Opens Right)

• Focus: (a, 0)
• Directrix: x = -a
• Vertex: (0, 0)
• Axis: y = 0 (x-axis)
• Length of Latus Rectum: 4a

1. y² = -4ax (Opens Left)

• Focus: (-a, 0)
• Directrix: x = a
• Vertex: (0, 0)
• Axis: y = 0 (x-axis)
• Length of Latus Rectum: 4a

1. x² = 4ay (Opens Upwards)

• Focus: (0, a)
• Directrix: y = -a
• Vertex: (0, 0)
• Axis: x = 0 (y-axis)
• Length of Latus Rectum: 4a

1. x² = -4ay (Opens Downwards)

• Focus: (0, -a)
• Directrix: y = a
• Vertex: (0, 0)
• Axis: x = 0 (y-axis)
• Length of Latus Rectum: 4a

Remember, a > 0 for all cases. The vertex is the turning point of the parabola.

The Ellipse (0 < e < 1)

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (the foci) is a constant. Its eccentricity 0 < e < 1. An ellipse has two axes of symmetry: the major axis (the longer axis passing through the foci) and the minor axis (the shorter axis perpendicular to the major axis).

Standard Equation of Ellipse:

1. x²/a² + y²/b² = 1 (Major axis along x-axis), where a > b

• Foci: (±c, 0), where c² = a² - b²
• Vertices: (±a, 0)
• Co-vertices: (0, ±b)
• Major Axis Length: 2a
• Minor Axis Length: 2b
• Eccentricity (e): c/a
• Directrices: x = ±a/e
• Length of Latus Rectum: 2b²/a

1. x²/b² + y²/a² = 1 (Major axis along y-axis), where a > b

• Foci: (0, ±c), where c² = a² - b²
• Vertices: (0, ±a)
• Co-vertices: (±b, 0)
• Major Axis Length: 2a
• Minor Axis Length: 2b
• Eccentricity (e): c/a
• Directrices: y = ±a/e
• Length of Latus Rectum: 2b²/a

Key relationship: c = ae. The centre of the ellipse is (0,0) in these standard forms.

The Hyperbola (e > 1)

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points (the foci) is a constant. Its eccentricity e > 1. Hyperbolas consist of two separate, open curves called branches.

Standard Equations of Hyperbola:

1. x²/a² - y²/b² = 1 (Transverse axis along x-axis)

• Foci: (±c, 0), where c² = a² + b²
• Vertices: (±a, 0)
• Transverse Axis Length: 2a (connecting vertices)
• Conjugate Axis Length: 2b (perpendicular to transverse axis)
• Eccentricity (e): c/a
• Directrices: x = ±a/e
• Length of Latus Rectum: 2b²/a
• Asymptotes: y = ±(b/a)x

1. y²/a² - x²/b² = 1 (Transverse axis along y-axis)

• Foci: (0, ±c), where c² = a² + b²
• Vertices: (0, ±a)
• Transverse Axis Length: 2a
• Conjugate Axis Length: 2b
• Eccentricity (e): c/a
• Directrices: y = ±a/e
• Length of Latus Rectum: 2b²/a
• Asymptotes: y = ±(a/b)x

Key relationship: c = ae. The centre of the hyperbola is (0,0) in these standard forms.

Worked Example

• {"title":"Identify the conic and find its focus, directrix, and latus rectum: y² = 12x","bodyMarkdown":"Solution:\n1. Identify: The equation y² = 12x is of the form y² = 4ax, which represents a parabola opening to the right.\n2. Find 'a': Comparing y² = 12x with y² = 4ax, we get 4a = 12, so a = 3.\n3. Focus: For y² = 4ax, the focus is (a, 0). Thus, the focus is (3, 0).\n4. Directrix: For y² = 4ax, the directrix is x = -a. Thus, the directrix is x = -3.\n5. Length of Latus Rectum: For a parabola, it's 4a. So, 4 * 3 = 12.\n\nAnswer: The conic is a parabola. Its focus is (3, 0), directrix is x = -3, and the length of the latus rectum is 12."}

Exam Tip: Identifying Conics from General Equation

For the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, primarily for Class 11, focus on equations without the xy term (i.e., B=0).

• If A=C (and not zero) and B=0: Circle
• If A and C have the same sign (and not zero) and A ≠ C (e.g., 2x² + 3y² = 1): Ellipse
• If A or C is zero (but not both), and B=0 (e.g., y² = 4x or x² = 2y): Parabola
• If A and C have opposite signs (e.g., x² - y² = 1): Hyperbola

Always bring the equation to its standard form to accurately identify properties. Pay close attention to signs and values of a and b (or c) as they define the orientation and size.

Key Points to Remember

• The eccentricity e is the most important characteristic defining a conic section (e=0 for circle, 0<e<1 for ellipse, e=1 for parabola, e>1 for hyperbola).
• For Parabola, y²=4ax opens right, y²=-4ax opens left, x²=4ay opens up, x²=-4ay opens down.
• For Ellipse, a is always associated with the major axis, b with the minor axis. c² = a² - b² and c = ae.
• For Hyperbola, a is associated with the transverse axis, b with the conjugate axis. c² = a² + b² and c = ae.
• The length of the latus rectum is 4a for parabola, 2b²/a for ellipse, and 2b²/a for hyperbola.
• Always remember the conditions a > b for ellipses and a can be greater than, less than, or equal to b for hyperbolas.
• The directrix is a line, and the focus is a point. Don't confuse their roles.
• Asymptotes are characteristic of hyperbolas and are lines that the branches approach as they extend indefinitely.

Practice Questions with Solutions

• Q: What is the eccentricity of a parabola? A: The eccentricity of a parabola is e = 1.
• Q: Find the focus of the parabola x² = 16y. A: Comparing with x² = 4ay, we have 4a = 16, so a = 4. The focus is (0, a), which is (0, 4).
• Q: For an ellipse x²/25 + y²/9 = 1, what are the lengths of its major and minor axes? A: Here a² = 25 and b² = 9, so a = 5 and b = 3. Major axis length 2a = 10. Minor axis length 2b = 6.
• Q: What is the relation between a, b, and c for a hyperbola? A: For a hyperbola, the relation is c² = a² + b².

Frequently Asked Questions