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Question

12. Foci(+ 3/5, 0), the latus rectum is of length 8.OCi

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Solution

The given coordinates of foci are ( ±3 5 ,0 ) .and length of latus rectum is 8.

Since the foci are on the x axis, the equation of the hyperbola is represented as,

x 2 a 2 y 2 b 2 =1 , where x is the transverse axis.(1)

Since x axis is the transverse axis, coordinates of Foci = (±c,0)

c=3 5 Length of latus rectum = 2 b 2 a

So, 2 b 2 a =8 b 2 =4a

a 2 + b 2 = c 2 a 2 +4a= ( 3 5 ) 2 ( a ) 2 +4a45=0 a 2 +9a5a45=0 ( a+9 )( a5 )=0

Hence, a=9,5

Since a is non-negative, a=5

Now, b 2 =4a b 2 =4×5 b= 20

Substitute the values of a and b in equation (1)

x 2 25 y 2 20 =1

Thus, the equation of hyperbola with foci ( ±3 5 ,0 ) and length of latus rectum is x 2 25 y 2 20 =1 .


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