Class 11 — Mathematics (NCERT)

Conic Sections

Class 11

✓By the end of this lesson students will be able to define conic sections as the locus of points satisfying specific geometric conditions.
✓By the end of this lesson students will be able to derive and apply the standard equations of a circle, parabola, ellipse, and hyperbola.
✓By the end of this lesson students will be able to identify and calculate key features of conic sections such as foci, vertices, directrix, eccentricity, and latus rectum.
✓By the end of this lesson students will be able to solve problems involving the properties and equations of conic sections.
✓By the end of this lesson students will be able to distinguish between different types of conic sections based on their equations and geometric properties.

Key concepts

Introduction to Conic Sections

A conic section is the curve obtained by the intersection of a plane with a double-napped right circular cone. The type of conic section formed depends on the angle at which the plane intersects the cone. The main types are circle, parabola, ellipse, and hyperbola. Degenerate conics (a point, a line, or a pair of intersecting lines) are formed when the plane passes through the vertex of the cone.

Circle

A circle is the locus of a point in a plane that moves in such a way that its distance from a fixed point (called the centre) is always constant. This constant distance is called the radius of the circle.

Standard equation of a circle with centre (h, k) and radius r: (x - h)² + (y - k)² = r²\nGeneral equation of a circle: x² + y² + 2gx + 2fy + c = 0, where centre is (-g, -f) and radius is √(g² + f² - c).

Parabola

A parabola is the locus of a point in a plane that moves in such a way that its distance from a fixed point (called the focus) is equal to its distance from a fixed line (called the directrix). The eccentricity (e) of a parabola is 1.

Standard equations of a parabola:\n1. y² = 4ax: Vertex (0,0), Focus (a,0), Directrix x = -a, Axis y = 0, Latus Rectum 4a.\n2. y² = -4ax: Vertex (0,0), Focus (-a,0), Directrix x = a, Axis y = 0, Latus Rectum 4a.\n3. x² = 4ay: Vertex (0,0), Focus (0,a), Directrix y = -a, Axis x = 0, Latus Rectum 4a.\n4. x² = -4ay: Vertex (0,0), Focus (0,-a), Directrix y = a, Axis x = 0, Latus Rectum 4a.

Ellipse

An ellipse is the locus of a point in a plane that moves in such a way that the sum of its distances from two fixed points (called foci) is constant. This constant sum is equal to the length of the major axis (2a). The eccentricity (e) of an ellipse is less than 1 (0 < e < 1).

Standard equations of an ellipse:\n1. x²/a² + y²/b² = 1 (a > b): Centre (0,0), Vertices (±a,0), Foci (±c,0) where c² = a² - b², Major axis 2a (along x-axis), Minor axis 2b (along y-axis), Eccentricity e = c/a, Latus Rectum 2b²/a.\n2. x²/b² + y²/a² = 1 (a > b): Centre (0,0), Vertices (0,±a), Foci (0,±c) where c² = a² - b², Major axis 2a (along y-axis), Minor axis 2b (along x-axis), Eccentricity e = c/a, Latus Rectum 2b²/a.

Hyperbola

A hyperbola is the locus of a point in a plane that moves in such a way that the absolute difference of its distances from two fixed points (called foci) is constant. This constant difference is equal to the length of the transverse axis (2a). The eccentricity (e) of a hyperbola is greater than 1 (e > 1).

Standard equations of a hyperbola:\n1. x²/a² - y²/b² = 1: Centre (0,0), Vertices (±a,0), Foci (±c,0) where c² = a² + b², Transverse axis 2a (along x-axis), Conjugate axis 2b (along y-axis), Eccentricity e = c/a, Latus Rectum 2b²/a.\n2. y²/a² - x²/b² = 1: Centre (0,0), Vertices (0,±a), Foci (0,±c) where c² = a² + b², Transverse axis 2a (along y-axis), Conjugate axis 2b (along x-axis), Eccentricity e = c/a, Latus Rectum 2b²/a.

Key facts to remember

1A conic section is formed by the intersection of a plane with a double-napped cone.
2The eccentricity (e) determines the type of conic: e=0 for circle, e=1 for parabola, 0<e<1 for ellipse, e>1 for hyperbola.
3The standard equation of a circle with centre (h,k) and radius r is (x-h)² + (y-k)² = r².
4For a parabola, every point is equidistant from the focus and the directrix.
5For an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant (2a).
6For a hyperbola, the absolute difference of the distances from any point on the hyperbola to the two foci is constant (2a).
7The latus rectum is a line segment perpendicular to the axis of a conic section, passing through a focus, and whose endpoints lie on the conic.
8The general equation of a conic section is Ax² + Bxy + Cy² + Dx + Ey + F = 0.

Worked examples

Example 1

Find the equation of the circle with centre (2, -3) and radius 5.

IThe standard equation of a circle with centre (h, k) and radius r is (x - h)² + (y - k)² = r².

IIGiven centre (h, k) = (2, -3) and radius r = 5.

IIISubstitute these values into the standard equation:

IV(x - 2)² + (y - (-3))² = 5²

V(x - 2)² + (y + 3)² = 25

VIExpand the terms:

VIIx² - 4x + 4 + y² + 6y + 9 = 25

VIIIx² + y² - 4x + 6y + 13 - 25 = 0

9x² + y² - 4x + 6y - 12 = 0

Answer

The equation of the circle is x² + y² - 4x + 6y - 12 = 0.

Remember to expand and simplify the equation to the general form if required.

Example 2

Find the coordinates of the focus, the equation of the directrix, the length of the latus rectum, and the equation of the axis of the parabola y² = 8x.

IThe given equation of the parabola is y² = 8x.

IIThis equation is of the form y² = 4ax.

IIIComparing y² = 8x with y² = 4ax, we get 4a = 8, which implies a = 2.

IVFor a parabola of the form y² = 4ax:

V1. Coordinates of the focus are (a, 0). Substituting a = 2, the focus is (2, 0).

VI2. Equation of the directrix is x = -a. Substituting a = 2, the directrix is x = -2.

VII3. Length of the latus rectum is 4a. Substituting a = 2, the length of the latus rectum is 4(2) = 8 units.

VIII4. Equation of the axis is y = 0.

Answer

Focus: (2, 0), Directrix: x = -2, Length of Latus Rectum: 8 units, Equation of Axis: y = 0.

Carefully identify the standard form of the parabola to correctly determine 'a' and then apply the corresponding formulas.

Example 3

Find the equation of the ellipse whose foci are (±5, 0) and the length of the major axis is 12.

IThe foci are given as (±5, 0). Since the foci are on the x-axis, the major axis is along the x-axis. The equation of the ellipse will be of the form x²/a² + y²/b² = 1.

IIFrom the foci (±c, 0), we have c = 5.

IIIThe length of the major axis is 2a = 12. Therefore, a = 12/2 = 6.

IVWe know that for an ellipse, c² = a² - b².

VSubstitute the values of a and c: 5² = 6² - b².

VI25 = 36 - b².

VIIb² = 36 - 25.

VIIIb² = 11.

9Now substitute the values of a² and b² into the standard equation x²/a² + y²/b² = 1.

10a² = 6² = 36.

11So, the equation of the ellipse is x²/36 + y²/11 = 1.

Answer

The equation of the ellipse is x²/36 + y²/11 = 1.

Always determine the orientation of the major axis first (x-axis or y-axis) based on the given information (foci or vertices) to choose the correct standard equation.

Common mistakes

✗Confusing the values of 'a' and 'b' in ellipse and hyperbola equations, especially when the major/transverse axis is along the y-axis.
✗Incorrectly identifying the sign of 'a' when determining the focus and directrix for parabolas opening left or downwards.
✗Mixing up the formulas for c² (c² = a² - b² for ellipse, c² = a² + b² for hyperbola).
✗Errors in algebraic manipulation, such as expanding squares or simplifying fractions, leading to incorrect final equations.
✗Not checking the orientation of the conic section (e.g., major axis along x or y) before applying the standard formulas for foci, vertices, etc.

Exam tips

★Memorise all standard forms of equations for each conic section along with their corresponding properties (foci, vertices, directrix, eccentricity, latus rectum).
★Always draw a rough sketch of the conic section when solving problems. This helps in visualising the orientation and properties.
★Pay close attention to the signs and coefficients in the given equations to correctly identify the type of conic and its parameters.
★Practice converting between general and standard forms of conic section equations to build proficiency in algebraic manipulation.

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