Ellipse is an important topic for the JEE exam. An ellipse is the set of all points on a plane whose distance from two points adds up to a constant. In this article, we have included the definition, ellipse equation and properties of an ellipse, which will help students to have a deep understanding of the topic. Some of the most important equations of an ellipse include area and circumference, tangent equation, tangent equation in slope form, chord equation, normal equation and the equation of chord joining the points of the ellipse.
What Is an Ellipse?
An ellipse is a locus of a point which moves in such a way that its distance from a fixed point (focus) to its perpendicular distance from a fixed straight line (directrix) is constant, i.e., eccentricity(e), which is less than unity.
(0 < e < 1)
Definitions of an Ellipse
Standard Equation of an Ellipse
The standard equation of an ellipse is given as:
(x^{2 }/ a^{2}) + (y^{2} / b^{2}) = 1 For, a>b
So, from the definition
[(SP) / (PM)] = e < 1 where P(x,y) is variable point.(x − ae)^{2} + (y − 0)^{2} = e^{2} [(x – a/e) / 1]^{2}
⇒ x^{2} (1 – e^{2}) + y^{2} = a^{2} (1 – e^{2})
Comparing both equations,
b^{2} = a^{2} (1 – e^{2}) e^{2 }= [(a^{2} – b^{2}) / a^{2}]
Where a ⇒ Semi major axis.
b ⇒ Semi minor axis.
Latus Rectum of an Ellipse
Chord LSL’ is called latus rectum
(x^{2 / }a^{2} + y^{2} / b^{2})= 1, substitute (x = ae)
(a^{2} e^{2} / a^{2}) + (y^{2} / b^{2}) = y^{2} = b^{2} (1 – e^{2}) y^{2} = (b^{4 }/ a^{2}) y = ± (b^{2 }/ a) ⇒ (b^{2 }/ a), (−b^{2 }/ a)
Length of latus rectum = 2b^{2}/a
Important Points to Remember:

Area of an ellipse
The area of an ellipse = πab, where a is the semi major axis and b is the semi minor axis.
Circumference of an ellipse
The approximate value of the circumference of an ellipse can be calculated as,
Position of point related to ellipse
Let the point p(x_{1}, y_{1}) and ellipse
(x^{2} / a^{2}) + (y^{2} / b^{2}) = 1
If [(x_{1}^{2 }/ a^{2})+ (y_{1}^{2} / b^{2}) − 1)]
= 0 {on the curve}
< 0 {inside the curve}
> 0 {outside the curve}
The parametric form of the equation of an ellipse
x = a cos θ
y = b sin θ
[(x^{2} / a^{2 })+ (y^{2} / b^{2})] = 1Auxiliary Circle
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Some Important Equations of an Ellipse
Some most important equations of an ellipse include tangent, tangent equation in slope form, chord equation, normal equation and the equation of chord joining the points of the ellipse. All these equations are explained below in detail.
Tangent of ellipse
The tangent of an ellipse is a line that touches a point on the curve of the ellipse.
Let the equation of ellipse be [(x^{2} / a^{2}) + (y^{2} / b^{2})] = 1
The slope at point p(x_{1}, y_{1})
Equation of tangent
y – y_{1} = [(− b^{2} x_{1} ) / (a^{2} y_{1})] (x – x_{1})
⇒ (xx_{1} / a^{2} ) + (yy_{1} / b^{2}) = 1, point form
⇒ at point (a cos θ, b sin θ), the equation of tangent is (x cos θ / a) + (y sin θ / b) = 1
⇒ Tangent in slope form y = mx + c
Where,
Equation of tangent in slope form
Tangent from an external point p(x_{1}, y_{1}) to the ellipse
(x^{2} / a^{2}) + (y^{2} / b^{2}) = 1.
SS_{1} = T^{2}
S ≡ (x^{2} / a^{2}) + (y^{2} / b^{2}) −1=0
S1 ≡ [(x_{1}^{2} / a^{2}) + (y_{1}^{2} / b^{2})] – 1 = 0
T ≡ [(xx_{1} / a^{2}) + (yy_{1} / b^{2})] – 1 = 0
Equation of chord with midpoint (x_{1}, y_{1})
The chord of an ellipse is a straight line which passes through two points on the ellipse’s curve. The chord equation of an ellipse having the midpoint as x_{1} and y_{1} will be:
T = S_{1}
(xx_{1} / a^{2}) + (yy_{1} / b^{2}) = (x_{1}^{2} / a^{2}) + (y_{1}^{2} / b^{2})
Equation of normal to an ellipse
The normal to an ellipse bisects the angle between the lines to the foci. The equation of the normal to an ellipse is:
Normal at point p (x_{1}, y_{1})
[(x – x_{1}) / (x_{1} /a^{2})] = [(y – y_{1}) / (y_{1} / b^{2})]Normal at point p (a cos θ, b sin θ)
The equation of chord joining the points of an ellipse
(a cos α, b sin α) and (a cos β, b sin β) can be given by:
If eccentric angles are α and β of the end of focal chords of the ellipse:
Then,
Ellipse and Hyperbola Important JEE Main Questions
Solved Problems on Ellipse
Problem 1: The line lx + my + n = 0 is a normal to the ellipse
Solution:
The equation of any normal to
The straight line lx + my + n = 0 …..(ii) will be a normal to the ellipse
Problem 2: If the normal at the point P(θ) to the ellipse
Solution:
The normal at P(a cos θ, b sin θ) is ax secθ − by cosecθ = a^{2} − b^{2} where a^{2} = 14, b^{2} = 5.
It meets the curve again at Q(2θ), i.e., (a cos 2θ, b sin 2θ)
Problem 3: The eccentricity of the conic 4x^{2} + 16y^{2} – 24x – 3y = 1 is
Solution:
Given the equation of conic is
Eccentric Angle and Auxiliary Circle of an Ellipse
Ellipse and Hyperbola – Important Topics
Ellipse and Hyperbola – Important Questions
Frequently Asked Questions
What do you mean by an ellipse?
An ellipse is the set of all points on a plane whose distance from two points adds up to a constant.
Give the standard equation of an ellipse.
The standard equation of an ellipse is given by (x^{2}/a^{2}) + (y^{2}/b^{2}) = 1. Here, a is the length of the semimajor axis, and b is the length of the semiminor axis.
Give the formula for the latus rectum of an ellipse.
The length of the latus rectum of an ellipse = 2b^{2}/a.
What is the eccentricity of an ellipse?
The eccentricity of the ellipse is less than 1. It is given by the formula, e = √(1 – (b^{2}/a^{2})).
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