find the coefficient of x⁷ in ( ax² + 1/bx)¹¹ and the coefficient of x^-7 in (ax - 1/bx²)¹¹ . if these coefficients are equal then find the relation between a and b

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$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} {\left(\mathrm{r}+1\right)}^{\mathrm{th}} \mathrm{term} \mathrm{in} \mathrm{expansion} \mathrm{of} {\left(\mathrm{x}+\mathrm{a}\right)}^{\mathrm{n}} \mathrm{is} \mathrm{given} \mathrm{by}: \\ {\mathrm{T}}_{\mathrm{r}+1}{=}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}.{\mathrm{x}}^{\mathrm{n}-\mathrm{r}}.{\mathrm{a}}^{\mathrm{r}} \\ \mathrm{We} \mathrm{have} {\left({\mathrm{ax}}^2+\frac{1}{\mathrm{bx}}\right)}^{11} \\ \mathrm{Let} {\mathrm{x}}^7 \mathrm{occur} \mathrm{in} {\left(\mathrm{r}+1\right)}^{\mathrm{th}} \mathrm{term}, \mathrm{hence} \\ {\mathrm{T}}_{\mathrm{r}+1}{=}^{11}{\mathrm{C}}_{\mathrm{r}}{\left({\mathrm{ax}}^2\right)}^{11-\mathrm{r}}{\left(\frac{1}{\mathrm{bx}}\right)}^{\mathrm{r}} \\ {=}^{11}{\mathrm{C}}_{\mathrm{r}}\left(\frac{{\mathrm{a}}^{11-\mathrm{r}}}{{\mathrm{b}}^{\mathrm{r}}}\right)\left(\frac{{\mathrm{x}}^{22-2\mathrm{r}}}{{\mathrm{x}}^{\mathrm{r}}}\right) \\ {=}^{11}{\mathrm{C}}_{\mathrm{r}}\left(\frac{{\mathrm{a}}^{11-\mathrm{r}}}{{\mathrm{b}}^{\mathrm{r}}}\right)\left({\mathrm{x}}^{22-2\mathrm{r}-\mathrm{r}}\right) \\ {=}^{11}{\mathrm{C}}_{\mathrm{r}}\left(\frac{{\mathrm{a}}^{11-\mathrm{r}}}{{\mathrm{b}}^{\mathrm{r}}}\right)\left({\mathrm{x}}^{22-3\mathrm{r}}\right) \\ \mathrm{As} \mathrm{this} \mathrm{term} \mathrm{contains} {\mathrm{x}}^7 \\ \Rightarrow 22-3\mathrm{r}=7 \\ \Rightarrow 3\mathrm{r}=15 \\ \Rightarrow \mathrm{r}=5 \\ \mathrm{Hence} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^7 \mathrm{in} {\left({\mathrm{ax}}^2+\frac{1}{\mathrm{bx}}\right)}^{11} \\ \mathrm{Coefficient} \mathrm{of} {\mathrm{x}}^7 {=}^{11}{\mathrm{C}}_5\left(\frac{{\mathrm{a}}^{11-5}}{{\mathrm{b}}^5}\right) \\ \Rightarrow \boxed{\mathbf{Coefficient}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{ }{\mathbf{x}}^{\mathbf{7}}{\mathbf{=}}^{\mathbf{11}}{\mathbf{C}}_{\mathbf{5}}\left(\frac{{\mathbf{a}}^{\mathbf{6}}}{{\mathbf{b}}^{\mathbf{5}}}\right)} \\ {\left(\mathrm{ax}-\frac{1}{{\mathrm{bx}}^2}\right)}^{11} \\ \mathrm{Let} {\mathrm{x}}^{-7} \mathrm{occurs} \mathrm{in} {\left(\mathrm{r}+1\right)}^{\mathrm{th}} \mathrm{term}, \mathrm{hence} \\ {\mathrm{T}}_{\mathrm{r}+1}{=}^{11}{\mathrm{C}}_{\mathrm{r}} {\left(\mathrm{ax}\right)}^{11-\mathrm{r}}{\left(-\frac{1}{{\mathrm{bx}}^2}\right)}^{\mathrm{r}} \\ ={\left(-1\right)}^{\mathrm{r}}{ }^{11}{\mathrm{C}}_{\mathrm{r}} \left(\frac{{\mathrm{a}}^{11-\mathrm{r}}}{{\mathrm{b}}^{\mathrm{r}}}\right)\left(\frac{{\mathrm{x}}^{11-\mathrm{r}}}{{\mathrm{x}}^{2\mathrm{r}}}\right) \\ ={\left(-1\right)}^{\mathrm{r}}{ }^{11}{\mathrm{C}}_{\mathrm{r}} \left(\frac{{\mathrm{a}}^{11-\mathrm{r}}}{{\mathrm{b}}^{\mathrm{r}}}\right)\left({\mathrm{x}}^{11-\mathrm{r}-2\mathrm{r}}\right) \\ ={\left(-1\right)}^{\mathrm{r}}{ }^{11}{\mathrm{C}}_{\mathrm{r}} \left(\frac{{\mathrm{a}}^{11-\mathrm{r}}}{{\mathrm{b}}^{\mathrm{r}}}\right)\left({\mathrm{x}}^{11-3\mathrm{r}}\right) \\ \mathrm{As} \mathrm{this} \mathrm{term} \mathrm{contains} {\mathrm{x}}^{-7}, \mathrm{hence}, \\ \Rightarrow 11-3\mathrm{r}=-7 \\ \Rightarrow \Rightarrow 3=18 \\ \Rightarrow \mathrm{r}=6 \\ \mathrm{Hence} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^{-7} \mathrm{in} {\left(\mathrm{ax}-\frac{1}{{\mathrm{bx}}^2}\right)}^{11}={\left(-1\right)}^6{ }^{11}{\mathrm{C}}_6 \left(\frac{{\mathrm{a}}^{11-6}}{{\mathrm{b}}^6}\right) \\ {\mathrm{C}}_2 \\ \mathrm{As}{ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}{=}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}-\mathrm{r}} \\ \Rightarrow \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^{-7} \mathrm{in} {\left(\mathrm{ax}-\frac{1}{{\mathrm{bx}}^2}\right)}^{11}{=}^{11}{\mathrm{C}}_{11-6} \left(\frac{{\mathrm{a}}^5}{{\mathrm{b}}^6}\right) \\ \Rightarrow \boxed{\mathbf{coefficient}\mathbf{ }\mathbf{of}\mathbf{ }{\mathbf{x}}^{\mathbf{-}\mathbf{7}}\mathbf{ }\mathbf{in}\mathbf{ }{\left(\mathbf{ax}\mathbf{-}\frac{\mathbf{1}}{{\mathbf{bx}}^{\mathbf{2}}}\right)}^{\mathbf{11}}{\mathbf{=}}^{\mathbf{11}}{\mathbf{C}}_{\mathbf{5}}\mathbf{ }\left(\frac{{\mathbf{a}}^{\mathbf{5}}}{{\mathbf{b}}^{\mathbf{6}}}\right)} \\ \mathrm{Given} \mathrm{that}: \\ \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^7 \mathrm{in} {\left({\mathrm{ax}}^2+\frac{1}{\mathrm{bx}}\right)}^{11} = \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^{-7} \mathrm{in} {\left(\mathrm{ax}-\frac{1}{{\mathrm{bx}}^2}\right)}^{11}{ \\ }^{11}{\mathrm{C}}_5\left(\frac{{\mathrm{a}}^6}{{\mathrm{b}}^5}\right){=}^{11}{\mathrm{C}}_5 \left(\frac{{\mathrm{a}}^5}{{\mathrm{b}}^6}\right) \\ \Rightarrow \frac{{\mathrm{a}}^6}{{\mathrm{b}}^5}=\frac{{\mathrm{a}}^5}{{\mathrm{b}}^6} \\ \Rightarrow \frac{{\mathrm{a}}^6}{{\mathrm{a}}^5}=\frac{{\mathrm{b}}^5}{{\mathrm{b}}^6} \\ \Rightarrow \mathrm{a}=\frac{1}{\mathrm{b}} \\ \Rightarrow \boxed{\mathbf{ }\mathbf{ab}\mathbf{=}\mathbf{1}\mathbf{ }} \end{gathered}$$

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