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 +4a−45=0 a 2 +9a−5a−45=0 ( a+9 )( a−5 )=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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