CBSE Class 11 Mathematics Conic Sections Notes Set 02

Important Notes and Formulas

Conic Sections

A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.
The fixed point is called the Focus.
The fixed straight line is called the Directrix.
The constant ratio is called the Eccentricity denoted by e.
The line passing through the focus & perpendicular to the directrix is called the Axis.
A point of intersection of a conic with its axis is called a Vertex.

Section of Right Circular Cone by Different Planes

A right circular cone is as shown in the fig. 1

(i) Section of a right circular cone by a plane passing through its vertex is a pair of straight lines passing through the vertex as shown in fig. 2.
(ii) Section of a right circular cone by a plane parallel to its base is a circle as shown in the fig. 3.
(iii) Section of a right circular cone by a plane parallel to a generator of the cone is a parabola as shown in the fig. 4.
(iv) Section of a right circular cone by a plane neither parallel to any generator of the cone nor perpendicular to the axis of the cone is an ellipse or hyperbola as shown in the fig. 5 & 6 (the plane meets only one nappe in an ellipse and both nappes in a hyperbola).

Parabola: Definition and Terminology

A parabola is the locus of a point, whose distance from a fixed point (focus) is equal to perpendicular distance from a fixed straight line (directrix).

Four standard forms of the parabola are
y² = 4ax; y² = −4ax; x² = 4ay; x² = −4ay
For parabola y² = 4ax:
(i) Vertex is (0, 0)    (ii) Focus is (a, 0)
(iii) Axis is y = 0    (iv) Directrix is x + a = 0
Focal Distance: The distance of a point on the parabola from the focus.
Focal Chord: A chord of the parabola, which passes through the focus.
Double Ordinate: A chord of the parabola perpendicular to the axis of the symmetry.
Latus Rectum: A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called the Latus Rectum (L.R.).
For y² = 4ax ⇒ Length of the latus rectum = 4a
⇒ ends of the latus rectum are L(a, 2a) & L′(a, −2a).

Note:
(i) Perpendicular distance from focus on directrix = half the latus rectum.
(ii) Vertex is middle point of the focus & the point of intersection of directrix & axis.
(iii) Two parabolas are said to be equal if they have the same latus rectum.

Ellipse: Standard Equation and Definitions

Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is
\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where a > b & b² = a²(1 − e²).

Eccentricity: \( e = \sqrt{1 - \frac{b^2}{a^2}} \), (0 < e < 1)
Foci: S ≡ (ae, 0) & S′ ≡ (−ae, 0)
Equations of Directrices: \( x = \frac{a}{e} \) & \( x = -\frac{a}{e} \)
Major Axis: The line segment A′A in which the foci S′ & S lie is of length 2a & is called the major axis (a > b) of the ellipse. Point of intersection of major axis with directrix is called the foot of the directrix (Z).
Minor Axis: The y-axis intersects the ellipse in the points B′ ≡ (0, −b) and B ≡ (0, b). The line segment B′B of length 2b (b < a) is called the minor axis of the ellipse.
Principal Axis: The major and minor axis together are called principal axis of the ellipse.
Vertices: A′ ≡ (−a, 0) & A ≡ (a, 0).
Focal Chord: A chord which passes through a focus is called a focal chord.
Double Ordinate: A chord perpendicular to the major axis is called a double ordinate.
Latus Rectum: The focal chord perpendicular to the major axis is called the latus rectum.
Length of latus rectum (LL′): \( \frac{2b^2}{a} = \frac{(\text{minor axis})^2}{\text{major axis}} = 2a(1 - e^2) \)
= 2e (distance from focus to the corresponding directrix).
Centre: The point which bisects every chord of the conic drawn through it is called the centre of the conic. C ≡ (0, 0), the origin is the centre of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

Note:
(i) If the equation of the ellipse is given as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and nothing is mentioned then the rule is to assume that a > b.
(ii) If b > a is given, then the y-axis will become major axis and x-axis will become the minor axis and all other points and lines will change accordingly.
(iii) If P be any point on the ellipse with S & S′ as foci then l(SP) + l(S′P) = 2a.

Hyperbola: Standard Equation and Definitions

The Hyperbola is a conic whose eccentricity is greater than unity (e > 1).

Standard equation of the hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where b² = a²(e² − 1).
Eccentricity (e): \( e^2 = 1 + \frac{b^2}{a^2} = 1 + \left(\frac{\text{C.A}}{\text{T.A}}\right)^2 \) (C.A → Conjugate Axis; T.A → Transverse Axis)
Foci: S ≡ (ae, 0) & S′ ≡ (−ae, 0).
Equations of Directrix: \( x = \frac{a}{e} \) & \( x = -\frac{a}{e} \)
Transverse Axis: The line segment A′A of length 2a in which the foci S′ & S both lie is called the transverse axis of the hyperbola.
Conjugate Axis: The line segment B′B between the two points B′ ≡ (0, −b) & B ≡ (0, b) is called as the conjugate axis of the hyperbola.
Principal Axes: The transverse & conjugate axis together are called Principal Axes of the hyperbola.
Vertices: A ≡ (a, 0) & A′ ≡ (−a, 0)
Focal Chord: A chord which passes through a focus is called a focal chord.
Double Ordinate: A chord perpendicular to the transverse axis is called a double ordinate.
Latus Rectum (l): The focal chord perpendicular to the transverse axis is called the latus rectum.
\( l = \frac{2b^2}{a} = \frac{(\text{C.A.})^2}{\text{T.A.}} = 2a(e^2 - 1) \)

Note:
(i) l (L.R.) = 2e (distance from focus to directrix).
(ii) Centre: The point which bisects every chord of the conic drawn through it is called the centre of the conic. C ≡ (0, 0), the origin is the centre of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
(iii) Since the fundamental equation to the hyperbola only differs from that to the ellipse in having −b² instead of b², it will be found that many propositions for the hyperbola are derived from those for the ellipse by simply changing the sign of b².

Solved Examples

Example 1. Find the lengths and equations of the focal radii drawn from the point \( (4\sqrt{3}, 5) \) on the ellipse 25x² + 16y² = 1600.
Answer:
Sol. The equation of the ellipse is 25x² + 16y² = 1600 or \( \frac{x^2}{64} + \frac{y^2}{100} = 1 \)

Here b > a: a² = 64, b² = 100
a² = b²(1 − e²) ∴ 64 = 100(1 − e²) ⇒ e = 3/5
Let \( P(x_1, y_1) \equiv (4\sqrt{3}, 5) \) be a point on the ellipse; then SP and S′P are the focal radii.
∴ \( SP = b - ey_1 \) and \( S'P = b + ey_1 \)
∴ \( SP = 10 - \frac{3}{5} \times 5 \) and \( S'P = 10 + \frac{3}{5} \times 5 \) ⇒ SP = 7 and S′P = 13
Also S is (0, be), i.e., \( \left(0, 10 \times \frac{3}{5}\right) \), i.e., (0, 6) and S′ is (0, −be), i.e., (0, −6).
∴ Equation of SP is \( y - 5 = \frac{6 - 5}{0 - 4\sqrt{3}}(x - 4\sqrt{3}) \)
⇒ \( -4\sqrt{3}y + 20\sqrt{3} = x - 4\sqrt{3} \) or \( x + 4\sqrt{3}y - 24\sqrt{3} = 0 \)
and equation of S′P is \( y - 5 = \frac{-6 - 5}{0 - 4\sqrt{3}}(x - 4\sqrt{3}) \)
⇒ \( -4\sqrt{3}y + 20\sqrt{3} = -11x + 44\sqrt{3} \) or \( 11x - 4\sqrt{3}y - 24\sqrt{3} = 0 \)

Example 2. If the angle between the straight lines joining foci and the end of minor axis of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is 90°, find its eccentricity.
Answer:
Sol. The equation of the ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
The ends of minor axis are B(0, b) and B′(0, −b). If the eccentricity of the ellipse is e, then the foci are S(ae, 0) and S′(−ae, 0).

∴ Slope of BS is \( m_1 = \frac{b - 0}{0 - ae} = -\frac{b}{ae} \) and slope of BS′ is \( m_2 = \frac{b - 0}{0 + ae} = \frac{b}{ae} \)
∴ The angle between BS and BS′ is 90°, ∴ m₁m₂ = −1 ⇒ \( -\frac{b}{ae} \times \frac{b}{ae} = -1 \)
⇒ b² = a²e² ⇒ a²(1 − e²) = a²e² ⇒ 1 − e² = e² ⇒ 2e² = 1
∴ \( e = \frac{1}{\sqrt{2}} \).

Example 3. Find the eccentricity of the hyperbola whose latus rectum is half of its transverse axis.
Answer:
Sol. Let the equation of hyperbola be \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
Then transverse axis = 2a and latus-rectum = \( \frac{2b^2}{a} \)
According to question \( \frac{2b^2}{a} = \frac{1}{2}(2a) \)
⇒ 2b² = a² (∵ b² = a²(e² − 1))
⇒ 2a²(e² − 1) = a² ⇒ 2e² − 2 = 1 ⇒ \( e^2 = \frac{3}{2} \)
∴ \( e = \sqrt{\frac{3}{2}} \)
Hence the required eccentricity is \( \sqrt{\frac{3}{2}} \).

Example 4. Obtain the equation of a hyperbola with co-ordinate axes as principal axes given that the distances of one of its vertices from the foci are 9 and 1 units.
Answer:
Sol. Let equation of hyperbola be \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) ...(1)
If vertices are A(a, 0) and A′(−a, 0) and foci are S(ae, 0) and S′(−ae, 0)
Given l(S′A) = 9 and l(SA) = 1 ⇒ a + ae = 9 and ae − a = 1
or a(1 + e) = 9 and a(e − 1) = 1
∴ \( \frac{a(1 + e)}{a(e - 1)} = \frac{9}{1} \) ⇒ 1 + e = 9e − 9 ⇒ \( e = \frac{5}{4} \)
∵ a(1 + e) = 9 ∴ \( a\left(1 + \frac{5}{4}\right) = 9 \) ⇒ a = 4
\( b^2 = a^2(e^2 - 1) = 16\left(\frac{25}{16} - 1\right) \) ∴ b² = 9
From (1) equation of hyperbola is \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \)