We will discuss about the latus rectum of the ellipse along with the examples.
Definition of the latus rectum of an ellipse:
The chord of the ellipse through its one focus and perpendicular to the major axis (or parallel to the directrix) is called the latus rectum of the ellipse.
It is a double ordinate passing through the focus. Suppose the equation of the ellipse be x2a2 + y2b2 = 1 then, from the above figure we observe that L1SL2 is the latus rectum and L1S is called the semi-latus rectum. Again we see that M1SM2 is also another latus rectum.
According to the diagram, the co-ordinates of the
end L1 of the latus
rectum L1SL2 are (ae,
SL1). As L1 lies on the ellipse x2a2 + y2b2 = 1, therefore, we
get,
(ae)2a2 + (SL1)2b2 = 1
a2e2a2 + (SL1)2b2 = 1
e2 + (SL1)2b2 =
1
⇒ (SL1)2b2 = 1 - e2
⇒ SL12 = b2 . b2a2, [Since, we know that, b2 = a2(1 - e2)]
⇒ SL12 = b4a2
Hence, SL1 = ± b2a.
Therefore, the co-ordinates of the ends L1 and L2 are (ae, b2a) and (ae, - b2a) respectively and the length of latus rectum = L1SL2 = 2 . SL1 = 2 . b2a = 2a(1 - e2)
Notes:
(i) The equations of the latera recta of the ellipse x2a2 + y2b2 = 1 are x = ± ae.
(ii) An ellipse has two latus rectum.
Solved examples to find the length of the latus rectum of an ellipse:
Find the length of the latus rectum and equation of the latus rectum of the ellipse x2 + 4y2 + 2x + 16y + 13 = 0.
Solution:
The given equation of the ellipse x2 + 4y2 + 2x + 16y + 13 = 0
Now form the above equation we get,
(x2 + 2x + 1) + 4(y2 + 4y + 4) = 4
⇒ (x + 1)2 + 4(y + 2)2 = 4.
Now dividing both sides by 4
⇒ (x+1)24 + (y + 2)2 = 1.
⇒ (x+1)222+(y+2)212 ………………. (i)
Shifting the origin at (-1, -2) without rotating the coordinate axes and denoting the new coordinates with respect to the new axes by X and Y, we have
x = X - 1 and y = Y - 2 ………………. (ii)
Using these relations, equation (i) reduces to X222 + Y212 = 1 ………………. (iii)
This is of the form X2a2 + Y2b2 = 1, where a = 2 and b = 1.
Thus, the given equation represents an ellipse.
Clearly, a > b. So, the given equation represents an ellipse whose major and minor axes are along X and Y axes respectively.
Now fine the eccentricity of the ellipse:
We know that e = √1−b2a2 = √1−1222 = √1−14 = √32.
Therefore, the length of the latus rectum = 2b2a = 2∙(1)22 = 22 = 1.
The equations of the latus recta with respect to the new axes are X= ±ae
X = ± 2 ∙ √32
⇒ X = ± √3
Hence, the equations of the latus recta with respect to the old axes are
x = ±√3 – 1, [Putting X = ± √3 in (ii)]
i.e., x = √3 - 1 and x = -√3 – 1.
● The Ellipse
11 and 12 Grade Math
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