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Latus Rectum of the Ellipse

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 = 1b2a2 = 11222 = 114 = 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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