If the eccentricity and length of latus rectum of a hyperbola are 13 3 and 10 3 units respectively, then what is the length of the transverse axis?

Step-by-step solution

Length of latus rectum of a hyperbola is 2 b^2 a where a is the half of the distance between two vertex of the hyperbola. Latus rectum = 2 b^2 a = 10 3 or, b^2= 5 a 3 ...(i) In case of hyperbola, b ^2= a ^2 ( e ^2-1 ) ...(ii) Putting value of b^2 from equation (i) and e= 13 3 in equation (ii), 5 a 3 = a ^2 ( 13 9 -1 ) or, 5 a 3 = 4 a^2 9 4 a^2-15 a=0 or, a(4-15 a)=0 a 0 , hence, a= 15 4