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, \(\frac{SA}{AK}\) = e : 1
\(\frac{SA}{AK}\) = \(\frac{e}{1}\)
⇒ SA = e ∙ AK ...................... (i) and
\(\frac{SA'}{A'K}\) = e : 1
\(\frac{SA'}{A'K}\) = \(\frac{e}{1}\)
⇒ 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 = \(\frac{a}{e}\) ...................... (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 = \(\frac{a}{e}\) – x, [Since, CK = \(\frac{a}{e}\)] and
SN = CS - CN = ae - x, [Since, CS = ae]
Since the point P lies on the required ellipse, Therefore, by the definition we get,
\(\frac{SP}{PM}\) = e
⇒ SP = e ∙ PM
⇒ SP\(^{2}\) = e\(^{2}\) . PM\(^{2}\)
or (ae - x)\(^{2}\) + (y - 0)\(^{2}\) = e\(^{2}\)[\(\frac{a}{e}\) - x]\(^{2}\)
⇒ x\(^{2}\)(1 – e\(^{2}\)) + y\(^{2}\) = a\(^{2}\)(1 – e\(^{2}\))
⇒ \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{a^{2}(1 - e^{2})}\) = 1
⇒ \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{a^{2}(1 - e^{2})}\) = 1
Since
0 < e < 1, hence a\(^{2}\)(1 - e\(^{2}\)) is always positive;
therefore, if a\(^{2}\)(1 - e\(^{2}\))
= b\(^{2}\), the above equation becomes, \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1.
The relation \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 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 \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 is called the standard equation of the ellipse.
Notes:
(i) b\(^{2}\) < a\(^{2}\), since e\(^{2}\) < 1 and b\(^{2}\) = a\(^{2}\)(1 - e\(^{2}\))
(ii) b\(^{2}\) = a\(^{2}\)(1 – e\(^{2}\))
⇒ \(\frac{b^{2}}{a^{2}}\) = 1 – e\(^{2}\), [Dividing both sides by a\(^{2}\)]
⇒ e\(^{2}\) = 1 - \(\frac{b^{2}}{a^{2}}\)
⇒ e = \(\sqrt{ 1 - \frac{b^{2}}{a^{2}}}\), [taking square root on both sides]
Form the above relation e = \(\sqrt{ 1 - \frac{b^{2}}{a^{2}}}\), we can find the value of e when a and b are given.
● The Ellipse
11 and 12 Grade Math
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