The hyperbola x^2 a^2 - y^2 b^2 =1 passes through the point (3 5 , 1) and the length of its latus rectum is 4 3 units. The length of the conjugate axis is

Step-by-step solution

x^2 a^2 - y^2 b^2 =1 Hyperbola passes through (3 5 , 1) aligned & (3 5 )^2 a ^2 - 1 ~b ^2 =1 & 45 a ^2 - 1 ~b ^2 =1...(i) aligned aligned & Now length of latus rectum = 2 b^2 a & 4 3 = 2 b^2 a & 2 3 = b ^2 a a = 3 ~b ^2 2 ...(ii) aligned Putting the value of ‘a’ from equation (ii) in equation (i), aligned & 45 4 9 b^4 - 1 b^2 =1 & 20 b^4 - 1 b^2 =1 aligned aligned & 20-b^2=b^4 & b^4+b^2-20=0 & b^4+5 b^2-4 b^2-20=0 & b^2 (b^2+5 )-4 (b^2+5 )=0 & (b^2-4 ) (b^2+5 )=0 & b^2=4, b^2=-5 & b^2=4 b=2 aligned Now length of co