Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
The standard equation of ellipse with reference to its principal axis along the coordinate axis is given by x^{2}/a^{2} + y^{2}/b^{2} = 1
In the standard equation, a > b and b^{2} = a^{2} (1 – e^{2}) Hence, the relation between a and b is a^{2} – b^{2} = a^{2}e^{2}, 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 2b^{2}/a = 2a(1 - e^{2})
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(x_{1}, y_{1}) lies outside, inside or on the ellipse according as x_{1}^{2}/a^{2} + y_{1}^{2}/b^{2} – 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
x^{2}/a^{2} + y^{2}/b^{2} = 1
a > b
a < b
centre
(0, 0)
vertex
(±a, 0)
(0, ±b)
Length of major axis
2a
2b
Length of minor axis
foci
(± ae, 0)
(0, ± be)
Equation of directirx
x = ± a/e
y = ± b/e
Relation between a, b and c
b^{2} = a^{2}(1 – e^{2})
a^{2} = b^{2}(1 – e^{2})
Equation of major axis
y = 0
x = 0
Equation of minor axis
Length of latus rectum
2b^{2}/a
2a^{2}/b
Ends of latus rectum
(± ae, ± b^{2}/a)
(± a^{2}/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 x^{2}/a^{2} + y^{2}/b^{2} = 1 in either two real, coincident or imaginary points according to whether c^{2} is < = or > a^{2}m^{2} + b^{2}
The equation y = mx + c is a tangent to the ellipse if c^{2} = a^{2}m^{2} + b^{2}
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 (x_{1}, y_{1}) is given by xx_{1}/a^{2} + yy_{1}/b^{2} = 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 (x_{1},y_{1}) is a^{2}x/x_{1} – b^{2}y/y_{1} = a^{2 }- b^{2} = a^{2}e^{2}
Equation of normal at the point (a cos θ a, b sin θ) is ax sec θ – by cosec θ = (a^{2} - b^{2})
Equation of normal in terms of its slope ‘m’ is y = mx – [(a^{2} - b^{2})m /√a^{2} + b^{2}m^{2}]
The equation of director circle is x^{2 }+ y^{2 }= a^{2} + b^{2}
Chord of contact:
Pair of tangents drawn form outside point P(x_{1}, y_{1}) to the ellipse meet it at R and P. Line joining P and R is called the chord of contact of point P(x_{1}, y1) with respect to the ellipse. The equation of chord of contact is
xx_{1}/a^{2} + yy_{1}/b^{2} = 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(x_{1}, y_{1}). Then the equation of the chord AB is T = S_{1}, where
S_{1} = x_{1}^{2}/a^{2} + y_{1}^{2}/b^{2} – 1 and T = xx_{1}/a^{2} + yy_{1}/b^{2} – 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.