If the coefficients of the middle terms in the binomial expansions of $(1 + \alpha x)^{26}$ and $(1

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If the coefficients of the middle terms in the binomial expansions of $(1 + \alpha x)^{26}$ and $(1 - \alpha x)^{28}$, $\alpha 
eq 0$, are equal, then the value of $\alpha$ is:

Accepted Answer

Sol. [Key Concept: The middle terms are $T_{14}$ of each expansion: set $26C_{13}\alpha^{13} = {}^{28}C_{14}\alpha^{14}$ and solve for $\alpha$.]$$(1 + \alpha x)^{26} $$Middle term, $T_{14} = \ ^{26}C_{13}\alpha^{13}$For $(1 - \alpha x)^{28}$Middle term $= \ ^{28}C_{14}\alpha^{14}$$${\Rightarrow \ ^{26}C_{13}\alpha^{13} = \ ^{28}C_{14}\alpha^{14} }{\Rightarrow \ ^{26}C_{13} = \ ^{28}C_{14}\alpha }{\Rightarrow \ ^{26}C_{13} = \frac{28}{14} 	imes \frac{27}{14}.\ ^{26}C_{13}\alpha }{\Rightarrow \alpha = \frac{7}{27} }$$

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