If ⁻¹ x- ⁻¹ y 3 = , where -1 x 1 , -3 y 3, x y 3 , then for all x, y9 x^2-6 x y +y^2 is equal to

Step-by-step solution

aligned & ⁻¹ a- ⁻¹ b= ⁻¹ (a b+ 1-a^2 1-b^2 ) & ⁻¹ x- ⁻¹ y 3 & = ⁻¹ ( x y 3 + 1-x^2 1- y^2 9 )= & x y 3 + 1-x^2 9-y^2 3 = & x y+ 1-x^2 9-y^2 =3 & x y-3 =- 1-x^2 9-y^2 aligned squaring on both sides, we get aligned & x^2 y^2-6 x y +9 ^2 = (1-x^2 ) (9-y^2 ) & x^2 y^2-6 x y +9 ^2 =9-y^2-9 x^2+x^2 y^2 aligned i.e., 9 x^2-6 x y +y^2=9 ^2