If the eccentricity e of the hyperbola x^2 a^2 - y^2 b^2 = 1 , passing through (6, 4 3 ) , satisfies 15(e^2 + 1) = 34e , then the length of the latus rectum of the hyperbola x^2 b^2 - y^2 2(a^2+1) = 1 is:

Step-by-step solution

The given equation of the hyperbola is x^2 a^2 - y^2 b^2 = 1 . Since it passes through (6, 4 3 ) , we have: 36 a^2 - 48 b^2 = 1 The eccentricity e satisfies 15(e^2 + 1) = 34e . 15e^2 - 34e + 15 = 0 (3e - 5)(5e - 3) = 0 Since the eccentricity of a hyperbola is e > 1 , we get e = 5 3 . We know that b^2 = a^2(e^2 - 1) . b^2 = a^2 ( 25 9 - 1 ) = 16 9 a^2 Substituting b^2 into the first equation: 36 a^2 - 48 16 9 a^2 = 1 36 a^2 - 27 a^2 = 1 9 a^2 = 1 a^2 = 9 Then, b^2 = 16 9 9 = 16 . The second hyperbola is given by x^2