If ⁻¹ x a - ⁻¹ x b = ⁻¹ b ⁻¹ a , then x is equal to

Step-by-step solution

⁻¹ ( x a )- ⁻¹ ( x b )= ⁻¹ b- ⁻¹ a ⁻¹ ( a x )- ⁻¹ ( b x )= ⁻¹ ( 1 b )- ⁻¹ ( 1 a ) ⁻¹ a b x^2 + 1- a^2 x^2 1- b^2 x^2 = ⁻¹ 1 a b + 1- 1 b^2 1- 1 a^2 [ ⁻¹ x- ⁻¹ y= ⁻¹ x y+ 1-x^2 1-y^2 ] . aligned & a b x^2 + (x^2-a^2 ) (x^2-b^2 ) x^2 = & 1 a b + (b^2-1 ) (a^2-1 ) a b aligned a^2 b^2+a b (x^2-a^2 ) (x^2-b^2 ) =x^2+x^2 (a^2-1 ) (b^2-1 ) a b (x^2-a^2 ) (x^2-b^2 ) -x^2 (a^2-1 ) (b^2-1 ) =x^2-a^2 b^2 in RHS x^2-a^2 b^2 has the minimum value zero because x^2 is always positive and a^2 b^2 is positive. If x^2-a^2 b^2=0 alig