Given that x = 1 6 is a solution to the equation ⁻¹(ax) + ⁻¹(bx) = 4 , where a and b are positive integers with a < b . The value of a+b is equal to

Step-by-step solution

Substitute x = 1 6 into the given equation: ⁻¹ ( a 6 ) + ⁻¹ ( b 6 ) = 4 Applying the inverse tangent addition formula: ⁻¹ ( a 6 + b 6 1 - ab 36 ) = 4 Taking tangent on both sides: 6(a+b) 36 - ab = 1 6a + 6b = 36 - ab ab + 6a + 6b = 36 Adding 36 to both sides to factorize: ab + 6a + 6b + 36 = 72 (a+6)(b+6) = 72 Since a and b are positive integers, a 1 and b 1 , which means a+6 7 and b+6 7 . The only pair of factors of 72 that are both greater than or equal to 7 is 8 and 9 . Since a a+6 = 8 a = 2 b+6 = 9 b = 3 Thus,