Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Elementary Properties of Ellipse

Definition:

It is such a locus of a point that moves in such a way that the ratio of its distance from a fixed point called focus and a fixed-line called directrix is constant, which is less than one. (Eccentricity = e < 1 )

Standard Equation of Ellipse:

Standard equation of ellipse to its principal axes along the co-ordinate axes is \dfrac{x^2}{a^2}+ \dfrac{y^2}{b^2}=1 , where   a > b  and  b^2 =a^2 (1-e^2)

\to Eccentricity (e)=\sqrt{1-\dfrac{b^2}{a^2}}, (0 < e < 1)

\to Focii :  F = (ae , 0)   and F'= (-ae ,0)

\to Equation of directrices :  x= \dfrac{a}{e}   and  x =-\dfrac{a}{e}

\to Minor Axis:

The  y-axis intersects the ellipse in the points B'= (0, -b)   and  B = (0, b) .

The line segment  B'B is of length 2b (b < a) is called the minor axis of the ellipse.

 \to Major Axis:

The line segment  A'A in which the focii  F and  F' lie is of length 2a and 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.

Principal Axis:

The major and minor axis together is called the principal axis of the ellipse.

Vertices:

Point of intersection of ellipse with the major axis A'= (-a ,0)   and  A =(a ,0)

Focal Chord:

A chord that 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  L.R  (L_1L_1')

 = \dfrac{2b^2}{a}

 =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  0 =(0, 0), is the origin of the centre of the ellipse  \dfrac{x^2}{a^2}+ \dfrac{y^2}{b^2}=1

Auxiliary Circle/ Eccentric Angle of Ellipse:

A circle described on the major axis of an ellipse as the diameter is called the auxiliary circle.

Hence  \theta= Eccentric angle of point P on the ellipse.

Parametric Representation of Ellipse:

The equation x =a \cos \theta   and y= b \sin \theta together represent the ellipse \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 where \theta is a parameter.

Note that if  P (\theta)=(a\cos \theta, b \sin \theta) is on the ellipse then;  Q (\theta)= (a\ocs \theta, a \sin \theta) is on the auxiliary circle.

The equation of chord of the ellipse joining two points with eccentric angles \alpha   and \beta is given by \dfrac{x}{a}\cos \dfrac{\alpha +\beta}{2}+\dfrac{y}{b}\sin \dfrac{\alpha +\beta}{2}= \cos \dfrac{\alpha -\beta}{2}

Example:

Equation of Ellipse \dfrac{x^2}{25}+\dfrac{y^2}{16}=1 joining two points  P\left( \dfrac{\pi}{4} \right) and Q\left( \dfrac{5\pi}{4} \right) .

Equation of chord is  \dfrac{x}{5}\cos \left( \dfrac{\frac{\pi}{4}+\frac{5\pi}{4}}{2} \right)+\dfrac{y}{4}\sin\left( \dfrac{\frac{\pi}{4}+\frac{5\pi}{4}}{2} \right)

=\cos \dfrac{\left( \frac{\pi}{4}-\frac{5\pi}{4} \right)}{2} \implies \dfrac{x}{5} \cos \dfrac{3\pi}{4}+\dfrac{y}{4}\sin \dfrac{3\pi}{4}=0

 \implies -\dfrac{x}{5}+\dfrac{y}{4}=0 \implies 4x=5y

Position of a Point w.r.t an Ellipse:

The point P(x_1 ,y_1) lies outside, inside or on the ellipse according as F_1  >  0, F_1 < 0 or F_1 =0 where  F_1 = \dfrac{x_1^2}{a^2}+\dfrac{y_1^2}{b^2} -1

Example:

Check whether the point P (1 ,-1) lies according to the ellipse  \dfrac{x^2}{25}+\dfrac{y^2}{16}=1

Solution:

 F_1 = \dfrac{1}{25}+\dfrac{1}{16}-1 <0

 \therefore P = (1 ,-1) lies inside the ellipse.

Line and an Ellipse:

The line  y=mx+c meets the ellipse  \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1  in two points real, coincident or imaginary according as  c^2 < a^2m^2+b^2, \quad c^2=a^2m^2+b^2 or   c^2 > a^2m^2+b^2

Tangent to Ellipse   \left( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \right)

(i) Slope form :  y=mx\pm \sqrt{a^2m^2+b^2} is tangent to the ellipse \left( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \right) , for all values of  m .

(ii) Point form : \left( \dfrac{xx_1}{a^2}+\dfrac{yy_1}{b^2}=1 \right) is tangent to ellipse  \left( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \right) at (x_1 , y_1)

(iii) Parametric form:\dfrac{x\cos \theta}{a}+\dfrac{y\sin \theta}{b}=1 is tangent to the ellipse \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 at the point (a\cos \theta, b \sin \theta)

Note:

\to There are two tangents to the ellipse having the same  m , i.e there are two tangents parallel to any given direction. These tangents touch the ellipse at extremities of a diameter.

 \to Point of intersection of the tangents at the point \alpha and \beta is \left(a \dfrac{\cos \dfrac{\alpha+\beta}{2}}{\cos \dfrac{\alpha-\beta}{2}}, b\dfrac{\sin\dfrac{\alpha+\beta}{2}}{\cos \dfrac{\alpha-\beta}{2}} \right)

 \to The eccentric angles of the points of contact of two parallel tangents differ by  \pi.

Normal to Ellipse:

(i) Equation of the normal at (x_1, y_1) to the ellipse \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 is \dfrac{a^2x}{x^1}-\dfrac{b^2y}{y_1}=a^2-b^2

(ii) Equation of the normal at the point (a\cos \theta, b \sin \theta) to the ellipse  \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 is ax\cdot \sec \theta -by\cdot \csc \theta=a^2-b^2

(iii) Equation of a normal in terms of its slope  'm' is y=mx-\dfrac{(a^2-b^2)m}{\sqrt{a^2+b^2m^2}}

Some Important Points:

(i) If  P be any point on the ellipse with focii  F and  F', then FP +F'P=2a

(ii) The tangent & normal at a point on the ellipse bisect the external & internal angles between the focal distances of this point.

(iii) The portion of the tangent to an ellipse between the point of contact & the directrix subtends a right angle at the corresponding focus.

(iv) The circle on any focal distance as diameter touches the auxiliary circle. Perpendiculars from the centre upon all the chords which join the ends of any perpendicular diameters of the ellipse are of constant length.

Application of Ellipse:

A whispering gallery is an elliptical-shaped room with a dome-shaped ceiling. If two people stand at the foci of the ellipse and whisper, they can hear each other, but others in this room cannot. “Statuary Hall in the U.S. Capital Building is a whispering gallery.”

Comparative Study of Two Ellipses

  Ellipse \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1, (a > b)  \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1, (b > a)
1 Centre  (0, 0) (0, 0)
2 Foci F (ae , 0), \quad F' (-ae, 0)  F (0, be), \quad F' (0, -be)
3 Directrices d=x = \dfrac{a}{e}, \quad d'=x= -\dfrac{a}{e} d=y = \dfrac{b}{e}, \quad d'=y= -\dfrac{b}{e}
4 Vertices  A (a, 0), \quad A'(-a, 0), \quad B (0, b), \quad B' (0, -b)  A (a, 0), \quad A'(-a, 0), \quad B (0, b), \quad B' (0, -b)
5 Axes

Major Axis ,  AA'=2a

Minor Axis, BB'=2b

Major Axis,  BB' =2b

MInor Axis, AA' =2a

6 Latus Rectum \dfrac{2b^2}{a} \dfrac{2a^2}{b}
7 Relation between  a, b and  e b^2=a^2 (1-e^2)  a^2=b^2(1-e^2)
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