If the coefficient of x^7 in [a x^2+ ( 1 b x ) ]¹¹ equals the coefficient of x⁻⁷ in [a x^2- ( 1 b x ) ]¹¹ , then a and b satisfy the relation

Step-by-step solution

aligned & T_ r+1 in the expansion [a x^2+ 1 b x ]¹¹= ¹¹ C_r (a x^2 )^ 11-r ( 1 b x )^r & = ¹¹ C_r(a)^ 11-r (b)^ -r (x)^ 22-2 r-r & 22-3 r=7 r=5 & coefficient of x^7= ¹¹ C₅(a)^6(b)⁻⁵ ....(i) aligned Again T_ r+1 in the expansion [a x- 1 b x^2 ]¹¹= ¹¹ C_r(a x)^ 11-r (- 1 b x^2 )^r= ¹¹ C _ r a ^ 11- r (-1)^ r ( b )^ - r ( x )^ -2 r ( x )^ 11- r Now 11-3 r=-7 3 r=18 r=6 aligned & coefficient of x⁻⁷= ¹¹ C₆ a^5 1 (b)-6 & ¹¹ C₅(a)^6(b)⁻⁵= ¹¹ C₆ a^5 (b)⁻⁶ & a b=1 aligned