Let the latus rectum of the hyperbola x 2 9 - y 2 b 2 = 1 subtend an angle of π 3 at the centre of the hyperbola. If b 2 is equal to l m ( 1 + n ) , where l and m are co-prime numbers, then l 2 + m 2 + n 2 is equal to __________.

Step-by-step solution

Given, Equation of hyperbola x 2 9 - y 2 b 2 = 1 And latusrectum L R subtends 60 ° at centre Now, plotting the diagram we get, Now, from above diagram we get A a e , b 2 a & B a e , - b 2 a ⇒ tan 30 ° = b 2 a a e = b 2 a 2 e = 1 3 ⇒ e = 3 b 2 9 as a 2 = 9 Also, e 2 = 1 + b 2 9 ⇒ 1 + b 2 9 = 3 b 4 81 ⇒ b 4 = 3 b 2 + 27 ⇒ b 4 - 3 b 2 - 27 = 0 ⇒ b 2 = 3 + 117 2 ignoring the negative sign as it is a square function ⇒ b 2 = 3 2 ( 1 + 13 ) Hence, on comparing with l m 1 + n we get, ⇒ l = 3 , m = 2 , n = 13 ⇒ l 2 + m 2 +