Let a, b > 0 . If the coefficient of x^8 in the expansion of (ax^2 + 1 2bx )¹⁰ is equal to the coefficient of x⁻⁸ in the expansion of (ax - 1 bx^2 )¹⁰ , then the term independent of x in the expansion of (ax + b x )^6 is

Step-by-step solution

For the expansion of (ax^2 + 1 2bx )¹⁰ , the general term is: T_ r+1 = ¹⁰C_ r (ax^2)^ 10-r ( 1 2bx )^r = ¹⁰C_ r a^ 10-r ( 1 2b )^r x^ 20-3r For the coefficient of x^8 , set 20 - 3r = 8 3r = 12 r = 4 . The coefficient is ¹⁰C₄ a^6 ( 1 2b )^4 = 210 a^6 16 b^4 . For the expansion of (ax - 1 bx^2 )¹⁰ , the general term is: T_ r+1 = ¹⁰C_ r (ax)^ 10-r (- 1 bx^2 )^r = ¹⁰C_ r a^ 10-r (- 1 b )^r x^ 10-3r For the coefficient of x⁻⁸ , set 10 - 3r = -8 3r = 18 r = 6 . The coefficient is ¹⁰C₆ a^4 (- 1 b )^6 = 210 a^4 b^6 . Equat