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Question

Equation of the latus rectum of the ellipse 9x2+4y2-18x-8y-23=0


A

y=±5

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B

y=-5

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C

y=1±5

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D

x=-1±5

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Solution

The correct option is C

y=1±5


Step 1. Solve the given equation:

9x2+4y218x8y23=0

(3x3)2+(2y2)21323=0

9(x1)2+4(y1)2=36

(x1)24+(y1)29=1

Step 2. Shifting origin to (1,1)

x1=X

y1=Y

X24+Y29=1

Here, a=2,b=3

The eccentricity of the ellipse, e2=1-a2b2

=1-49

=59

e=±59

=±53

The focus of the ellipse =(0,±be)

=(0,±5)

Step3: Find the latus rectum of the ellipse:

latus rectum will beY=±5

Shifting back, Y=y1

y1=±5

y=1±5

Thus, the equation of latus rectum is y=1±5

Hence, Option ‘C’ is Correct.


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