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Standard Equation of an Ellipse

We will learn how to find the standard equation of an ellipse.

Let S be the focus, ZK the straight line (directrix) of the ellipse and e (0 < e < 1) be its eccentricity. From S draw SK perpendicular to the directrix KZ. Suppose the line segment SK is divided internally at A and externally at A' (on KS produced) respectively in the ratio e : 1.

Therefore, SAAK = e : 1

SAAKe1

⇒ SA = e ∙ AK ...................... (i) and 

SAAK = e : 1

SAAK = e1

⇒ SA' = e ∙ A'K ...................... (ii)

We can clearly see that the points A and A'' lies on the ellipse since, their distance from the focus (S) bear constant ratio e (< 1) to their respective distance from the directrix.

Let C be the mid-point of the line-segment AA'; draw CY perpendicular to AA'.

Now, let us choose C as the origin CA and CY are chosen as x and y-axes respectively.

Therefore, AA' = 2a

A'C = CA = a.

Now, adding (i) and (ii) we get,

SA + SA' = e (AK + A'K) 

AA' = e (CK - CA + CK + CA')

2a = e (2CK - CA + CA')

2a = 2e CK,  (Since, CA = CA')

CK = ae ...................... (iii)

Similarly, subtracting (i) from (ii) we get,

SA' - SA = e (KA' - AK)

(CA' + CS) - (CA - CS) = e . (AA')

2CS = e 2a, [Since, CA' = CA]    

CS = ae ...................... (iv)

Let P (x, y) be any point on the required ellipse. From P draw PM perpendicular to KZ and PN perpendicular to CX and join SP.

Then, CN = x, PN = y and

PM = NK = CK - CN = ae – x, [Since, CK = ae] and

SN = CS - CN = ae - x, [Since, CS = ae]  

Since the point P lies on the required ellipse, Therefore, by the definition we get,

SPPM = e   

SP = e PM

SP2 = e2 . PM2

or  (ae - x)2 + (y - 0)2 = e2[ae - x]2

⇒ x2(1 – e2) + y2 = a2(1 – e2)

x2a2 + y2a2(1e2) = 1

x2a2 + y2a2(1e2) = 1

Since 0 < e < 1, hence a2(1 - e2) is always positive; therefore, if a2(1 - e2) = b2, the above equation becomes,  x2a2 + y2b2 = 1. 

The relation x2a2 + y2b2 = 1 is satisfied by the co-ordinates of all points P (x, y) on the required ellipse and hence, represents the required equation of the ellipse.

The equation of an ellipse in the form x2a2 + y2b2 = 1 is called the standard equation of the ellipse.


Notes:

(i) b2 < a2, since e2 < 1 and b2 = a2(1 - e2)

(ii)  b2 = a2(1 – e2)

b2a2 = 1 – e2, [Dividing both sides by a2]   

e2 = 1 - b2a2  

e = 1b2a2, [taking square root on both sides]

Form the above relation e = 1b2a2, we can find the value of e when a and b are given.

● The Ellipse


11 and 12 Grade Math 

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