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Standard Equations of an Ellipse

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Standard Equations of an Ellipse - Lesson Summary

An ellipse is the set of all the points in a plane the sum of whose distances from two fixed points is constant.

P1F1 + P1F2 = P2F1 + P2F2 = P3F1 + P3F2

Two foci, two vertices, centre, major axis and minor axis of an ellipse are as shown in the figure.

In an ellipse,

c2 = a2 - b2,

Where,
c = Distance of a focus from the centre
a = Length of semi-major axis
b = Length of semi-minor axis

P1F1 + P1F2 = P2F1 + P2F2 = P3F1 + P3F2 = k

Let K be the constant equal to the sum of the distances of any point on the ellipse from its foci.

Vertex A lies on the ellipse,

⇒ AF1 + AF2 = k ……(1)

AF1 = AO - F1O
⇒ AF1 = a - c

AF2 = AO + OF2
⇒ AF2 = a + c

⇒ AF1 + AF2 = a - c + a + c = k
⇒ k = 2a
P1F1 + P1F2 = P2F1 + P2F2 = P3F1 + P3F2 = 2a
Thus, the sum of the distances of any point on an ellipse from its foci is equal to the length of the major axis, or two times the semi-major axis of the ellipse.

Eccentricity is a measure of the deviation of a conic section from being a circle. The eccentricity of an ellipse is denoted by 'e' and is equal to the ratio of the distance of a focus from the centre to the length of the semi-major axis of the ellipse. Eccentricity of an ellipse (e) = c/a

Squaring both sides, we get

e2 = c2/a2

e2 = (a2 - b2)/a2 (Since c2 = a2 - b2)

e2 = 1- b2/a2

a2e2 = a2- b2

b2 = a2 - a2e2

b2 = a2 (1- e2)

Some standard equations that are satisfied by all the points lying on an ellipse

If we consider an ellipse with its vertex at the origin and its major axis along the X- or the Y-axis, then there are two distinct possibilities.

Case I: Centre (0, 0), major axis along the X-axis.

Case II: Centre (0, 0), major axis along the Y-axis.

Consider a point P on the ellipse with the coordinates X, Y.

PF1 + PF2 = 2a ……(1)

Distance between the points (x1, y1) and (x2, y2) = √(x2 - x1)2 + (y2 - y1)2

PF1 = √[(x - (-c)]2 + (y - 0)2

⇒ PF1 = √(x + c)2 + y2 ……(2)

PF2 = √(x - c)2 + (y - 0)2

⇒ PF2 = √(x - c)2 + y2 ……(3)

From equations (1), (2) and (3),

√(x + c)2 + y2 + √(x - c)2 + y2 = 2a

⇒ √(x + c)2 + y2 = 2a - √(x - c)2 + y2

Squaring both sides, we get

(x + c)2 + y2 = [2a - √(x - c)2 + y2]2

(x + c)2 + y2 = (2a)2 - 2(2a)(√(x - c)2 + y2) + (√(x - c)2 + y2)2

(x + c)2 + y2 = 4a2 - 4a√(x - c)2 + y2 + (x - c)2 + y2

⇒ 4a√(x - c)2 + y2 = (x - c)2 - (x + c)2 + 4a2

= x2 + c2 - 2xc -x2 - c2 - 2xc + 4a2

= 4a2 - 4xc

a√(x - c)2 + y2 = a2 - xc

a√(x - c)2 + y2 = a2 - xc

⇒ √(x - c)2 + y2 = a - (c/a) x

Squaring both sides, we get

(x - c)2 + y2 = [a - (c/a) x]2

(x - c)2 + y2 = (a)2 - 2.a.(c/a) x + ((c/a) x)2

(x - c)2 + y2 = a2 + (c2/a2) x - 2cx

x2 + c2 - 2cx + y2 = a2 + (c2/a2)x2 - 2cx

x2 + c2 + y2 = a2 + (c2/a2)x2

x2 - (c2/a2)x2 + y2 = a2 - c2

⇒(1 - (c2/a2))x2 + y2 = a2 - c2

⇒( (a2-c2)/a2)x2 + y2 = a2 - c2 ……(4)

⇒( b2/a2)x2 + y2 = b2 (Since a2 - c2 = b2)

Multiplying both sides by 1/b2,

x2/a2 + y2/b2 = 1 ……(5)

y 2 = b 2 (1 - x 2/a 2 (From equation (5))

⇒ PF1 = √(x + c)2 + b2 (1 - x2/a2 From equations ...(2) and (5))

= √ x2 + c2 + 2cx + b2 - b2x2/a2

= √ x2(1- b2/a2 + 2cx + c2 + b2

= √ x2((a2-b2)/a2) + 2cx + c2 + b2

= √(c2/a2)x2+ 2cx + a2 (Since c2 = a2 - b2)

= √(a + c/a)x2

⇒ PF1 = a + (c/a)x

⇒ PF2 = √(x - c)2 + b2 (1 - x2/a2) (From equation (3),

= √ x2 + c2 - 2cx + b2 - b2x2/a2

= √ x2(1- b2/a2 - 2cx + c2 + b2

= √ x2(a2-b2/a2)- 2cx + c2 + b2

= √(c2/a2)x2- 2cx + a2

= √(a - cx/a)2

⇒ PF2 = a - cx/a

⇒ PF1 + PF2 = a + cx/a + a - cx/a = 2a

The equation x2/a2 + y2/b2 = 1 is the standard equation for an ellipse with its centre lying at the origin and the major axis lying along the X-axis.

Similarly, the standard equation for an ellipse with its centre at the origin and the major axis along the Y-axis can be derived as x2/b2 + y2/a2 = 1.

These two equations are called the standard equations of an ellipse having its centre at the origin and the major axis along the X- or the Y-axis.

In an ellipse, the length of the major axis is always greater than the length of the minor axis.
If the denominator of the x2 term is greater than the denominator of the y2 term, then the major axis lies along the X-axis.
If the denominator of the y2 term is greater than the denominator of the x2 term, then the major axis lies along the Y-axis.

Both the standard equations of an ellipse contain even powers of X and Y.

An ellipse with its centre lying at the origin and the major axis along either the X- or the Y-axis is symmetrical about both the coordinate axes.

⇒ If (x, y) lies on the ellipse, then (-x, y), (x, -y) and (-x, -y) also lie on the ellipse.


Length of Latus Rectum

A line segment passing through the focus and perpendicular to the major axis with its end points lying on the curve of an ellipse is called the latus rectum of the ellipse.

Since an ellipse has two foci, it has two latus recta. Latus recta is the plural for latus rectum.

Consider an ellipse with its centre at the origin and the major axis along the X-axis and Latus Recta AB and CD.

Let AF2 = l

⇒ Coordinates of A are (c, l), where

c = Distance of focus F2 from centre O

Equation of the given ellipse is x2/a2 + y2/b2 = 1.

Sine A (c, l) lies on x2/a2 + y2/b2 = 1.

⇒ c2/a2 + l2/b2 = 1
⇒ l2 = b2( 1 - c2/a2 )
⇒ l2 = b2( (a2 - c2)/a2 )
⇒ l2 = b2( b2/a2)              (Since b2 = a2 - c2)
⇒ l2 = b4/a2

⇒ l = b2/a

AF2 = F2B =  b2/a

⇒ AB = AF2 + F2B = 2b2/a

Length of latus rectum of an ellipse = 2b2/a, where

a = Length of semi-major axis

b = Length of semi-minor axis.

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