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# Revision Notes on Ellipse

• The standard equation of ellipse with reference to its principal axis along the coordinate axis is given by x2/a2 + y2/b2 = 1

• In the standard equation, a > b and b2 = a2 (1 – e2) Hence, the relation between a and b is a2 – b2 = a2e2, where ‘e’ is the eccentricity and 0 < e < 1. • The foci of the ellipse are S(ae, 0) and S’ = (-ae, 0)

• Equations of the directrices are given by x = a/e and x = -a/e

• The coordinates of vertices are A’ = (-a, 0) and A = (a,0)

• The lengths of the major and minor axis are 2a and 2b respectively.

• The length of latus rectum is 2b2/a = 2a(1 - e2)

• Distance between the two foci is 2ae and distance between directrix is 2a/e.

• Two ellipses are said to be similar if they have the same eccentricity.

• The sum of the focal distances of any pint on the ellipse is equal to the major axis. As a result, the distance of focus from the extremity of a minor axis is equal to semi major axis.

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

• If a question does not mention the relation between a and b then by convention a is assumed to be greater than b i.e. a > b.

• The point P(x1, y1) lies outside, inside or on the ellipse according as x12/a2 + y12/b2 – 1 >, < or = 0.

• In parametric form, the equations x = a cos θ and y = b sin θ together represent the ellipse.

• Comparison Chart between Standard Ellipse:?

 Basic Elements x2/a2 + y2/b2 = 1 a > b a < b centre (0, 0) (0, 0) vertex (±a, 0) (0, ±b) Length of major axis 2a 2b Length of minor axis 2b 2a foci (± ae, 0) (0, ± be) Equation of directirx x = ± a/e y = ± b/e Relation between a, b and c b2 = a2(1 – e2) a2 = b2(1 – e2) Equation of major axis y = 0 x = 0 Equation of minor axis x = 0 y = 0 Length of latus rectum 2b2/a 2a2/b Ends of latus rectum (± ae, ± b2/a) (± a2/b, ± be) Distance between foci 2ae 2be Distance between directrix 2a/e 2b/e Parametric equation (a cos θ, b sin θ) (0 < θ < 2π) (a cos θ, b sin θ)
• The line y = mx + c meets the ellipse x2/a2 + y2/b2 = 1 in either two real, coincident or imaginary points according to whether c2 is < = or > a2m2 + b2

• The equation y = mx + c is a tangent to the ellipse if c2 = a2m2 + b2

• The equation of the chord of ellipse that joins two points with eccentric angles α and β is given by
x/a cos (α + β)/2 + y/b sin (α + β)/2 = cos (α - β)/2

• Equation of tangent to the ellipse:

• The equation of tangent to the ellipse at the point (x1, y1) is given by xx1/a2 + yy1/b2 = 1

• In parametric form, (x cos θ) /a + (y sin θ) /b = 1 is the tangent to the ellipse at the point (a cos θ, b sin θ).

• Equation of normal:

• Equation of normal at the point (x1,y1) is a2x/x1 – b2y/y1 = a- b2 = a2e2

• Equation of normal at the point (a cos θ a, b sin θ) is ax sec θ – by cosec θ = (a2 - b2)

• Equation of normal in terms of its slope ‘m’ is y = mx – [(a2 - b2)m /√a2 + b2m2]

• The equation of director circle is x+ y= a2 + b2

• Chord of contact:

Pair of tangents drawn form outside point P(x1, y1) to the ellipse meet it at R and P. Line joining P and R is called the chord of contact of point P(x1, y1) with respect to the ellipse. The equation of chord of contact is

xx1/a2 + yy1/b2 = 1. • The portion of the tangent to an ellipse between the point of contact and the directrix subtends a right angle at the corresponding focus.

• The perpendiculars from the center upon all chords which join the ends of any particular diameters of the ellipse are of constant length.

• Chord with a given middle point:

AB is a chord of the circle whose mid-point is P(x1, y1). Then the equation of the chord AB is T = S1, where

S1 = x12/a2 + y12/b2 – 1 and T = xx1/a2 + yy1/b2 – 1.

• Two diameters of ellipse are said to be conjugate diameters if each bisects the chords parallel to the other.

• The eccentric angles at the ends of a pair of conjugate diameters of an ellipse differ by a right angle.

• The sum of squares of any two conjugate semi-diameters of an ellipse is constant and equal to the sum of squares of semi-axis of ellipse.