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M14.2 – Ellipse
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M14.2 Ellipse
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M14.2 – Ellipse
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M14.2 Ellipse
LEARNING OBJECTIVES
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Derive the mathematical expression and geometrical representation of a standard ellipse based on its definition.
Define important terms such as eccentricity, focus, directrix, major and minor axis, vertex, centre, double
ordinate, latus rectum, focal distance and focal chord of an ellipse and derive their values / expressions for a
standard ellipse
Define both the x and y co-ordinates of any point on an ellipse using a common parameter.
Find equation of tangents and normal to a standard ellipse using:
(a) Coordinates of a point on the ellipse (point form)
(b) The common parameter for a point lying on the ellipse (Parametric form)
(c) slope of a line (slope form)
Obtain equation of the pair of tangents to a standard ellipse from an external point, the equation of the
corresponding chord of contact, and the equation of a chord bisected at a point.
PRE- READING
Ellipse Definition
An ellipse is the locus of a point which moves in a plane such that the ratio of its
distance from a fixed point (i.e. focus) to its distance from a fixed line (directrix) is a
constant.(i.e. directrix). This ratio is called eccentricity and is denoted by 𝒆. For an
ellipse 0 < 𝑒 < 1.
An alternative definition is that an ellipse is the locus of a point which moves in such a way
that the sum of its distances from two fixed points is constant. These two fixed points are the
foci of the ellipse.
Standard equation of ellipse
Let 𝑆 be the focus and ZM the directrix of the ellipse. Draw 𝑆𝑍 ⊥ 𝑍𝑀. Divide SZ internally and externally in the ratio
𝑒: 1(𝑒 < 1) and let A and 𝐴′ be these internal and external points of division.
Then 𝑆𝐴 = 𝑒𝐴𝑍 …………. (1)
And 𝑆𝐴′ = 𝑒𝐴′𝑍 …………….. (2)
Clearly A and A’ will lie on the ellipse
Let 𝐴𝐴′ = 2𝑎 and take C, the midpoint of 𝐴𝐴′ as origin.
∴ 𝐶𝐴 = 𝐶𝐴′ = 𝑎
adding (1) and (2),
𝑆𝐴 + 𝑆𝐴′ = 𝑒(𝐴𝑍 + 𝐴′ 𝑍)
⇒ 𝐴𝐴′ = 𝑒(𝐶𝑍 − 𝐶𝐴 + 𝐶𝐴′ + 𝐶𝑍) (From figure)
⇒ 𝐴𝐴′ = 𝑒(2𝐶𝑍) (∵ CA=CA’)
⇒ 2𝑎 = 2𝑒𝐶𝑍
M14.2
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2014 - 2016
M14.2 – Ellipse
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∴ 𝐶𝑍 = 𝑎/𝑒
∴ The directrix MZ is 𝑥 = 𝐶𝑍 =
𝑎
𝑎
𝑒
𝑒
Or 𝑥 − = 0 (∵ 𝑒 < 1, ∴
𝑎
𝑒
> 𝑎)
And subtracting (1) from (2), then
𝑆𝐴′ − 𝑆𝐴 = 𝑒(𝐴′ 𝑍 − 𝐴𝑍)
⇒ (𝐶𝐴′ + 𝐶𝑆) − (𝐶𝐴 − 𝐶𝑆) = 𝑒(𝐴𝐴′ )
⇒ 2𝐶𝑆 = 𝑒(𝐴𝐴′ ) (∵ 𝐶𝐴 = 𝐶𝐴′ )
⇒ 2𝐶𝑆 = 𝑒(2𝑎)
∴ 𝐶𝑆 = 𝑎𝑒
∴ The focus S is (𝐶𝑆, 0)𝑖. 𝑒. , (𝑎𝑒, 0)
Let 𝑃(𝑥, 𝑦) be any point on the ellipse.
Now draw 𝑃𝑀 ⊥ 𝑀𝑍
∴
𝑆𝑃
=𝑒
𝑃𝑀
Or (𝑆𝑃)2 = 𝑒 2 (𝑃𝑀)2
𝑎 2
(𝑥 − 𝑎𝑒)2 + (𝑦 − 0)2 = 𝑒 2 (𝑥 − )
𝑒
⇒ (𝑥 − 𝑎𝑒)2 + 𝑦 2 = (𝑒𝑥 − 𝑎)2
⇒ 𝑥 2 + 𝑎2 𝑒 2 − 2𝑎𝑒𝑥 + 𝑦 2 = 𝑒 2 𝑥 2 − 2𝑎𝑒𝑥 + 𝑎2
⇒ 𝑥 2 (1 − 𝑒 2 ) + 𝑦 2 = 𝑎2 (1 − 𝑒 2 )
⇒
Or
𝑥2
𝑦2
+ 2
=1
2
𝑎
𝑎 (1 − 𝑒 2 )
𝑥2
𝑎2
+
𝑦2
𝑏2
= 1 where 𝑏 2 = 𝑎2 (1 − 𝑒 2 )
This is the standard equation of the ellipse.
Conclusion:
𝑥2
𝑎2
+
𝑦2
𝑏2
= 1 is the standard equation of an ellipse with eccentricity e = √1 −
(±𝑎𝑒, 0) and corresponding directrices at 𝑥 = ±
𝑏2
𝑎2
and a pair of foci at
𝑎
𝑒
Key Properties

An ellipse has 2 foci and 2 directrices.

The major axis of an ellipse is the chord joining the two foci. The length of the major axis is 2a for a standard
ellipse, and it coincides with the x axis or y = 0.

The minor axis of an ellipse is the chord passing through the centre and perpendicular to the major axis.
The length of the minor axis is 2b for a standard ellipse, and it coincides with the y axis or x = 0.
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2014 - 2016
M14.2
M14.2 – Ellipse
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2𝑎

Distance between foci = 2𝑎𝑒 and distance between directrices =

If e = 0, then 𝑏 2 = 𝑎2 and equation changes to that of a circle

The sum of focal distances of a point on the ellipse is constant and is equal to the length of the major axis
of the ellipse, i.e. PS+PS’=2a
𝑒
(for a standard ellipse)
M14.2
Avanti Learning Centres Pvt Ltd. All rights reserved.
2014 - 2016