Question

Find the coefficient of $x^7$ in ${(a{x}^2+\frac{1}{bx})}^{11}$ and that of $x^{-7}$ in ${(ax-\frac{1}{b{x}^2})}^{11}$ and then find the relation between a and b so that these coefficients are equal, none of a, b and x is zero.

Answer 1

Co-efficient of $x^7$ in $(a{x}^2+\frac{1}{bx}{)}^{11}$ = ${}^{11}{C}_r\frac{a^{11-r}}{b^r}{x}^{22-3r}$

Atp, $22-3r=7$

$r=5$

Co-efficient = ${}^{11}{C}_5\frac{a^6}{b^5}$

Similarly, that of $x^{-7}$ in $(ax-\frac{1}{b{x}^2}{)}^{11}{=}^{11}{C}_6\frac{a^5}{b^6}$

If they are equal,

${}^{11}{C}_5\frac{a^6}{b^5}{=}^{11}{C}_6\frac{a^5}{b^6}$

$\frac{a^6}{b^5}=\frac{a^5}{b^6}$

$ab=1$