Question

13.Foci仕4,0), the latus rectum is of length 12

Answer 1

The given coordinates of foci are ( ±4,0 ) .and length of latus rectum is 12.

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=4

Length of latus rectum = 2 b 2 a

So, 2 b 2 a =12 b 2 a =6 b 2 =6a

a 2 + b 2 = c 2 a 2 +6a=16 a 2 +8a−2a−16=0 ( a+8 )(a−2)=0 a=−8,2

Since a is non-negative, a=2

Now, b 2 =6a b 2 =6×2 b= 12

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

x 2 4 − y 2 12 =1

Thus, the equation of hyperbola with foci ( ±4,0 ) and length of latus rectum 12 is x 2 4 − y 2 12 =1 .