If the coefficient of x 8 in ( a x 2 + 1 b x ) 13 is equal to the coefficient of x - 8 in ( a x - 1 b x 2 ) 13 , then a and b will satisfy the relation

Question

If the coefficient of $x^{8}$ in ${(a x^{2} + \frac{1}{b x})}^{13}$ is equal to the coefficient of $x^{- 8}$ in ${(a x - \frac{1}{b x^{2}})}^{13},$ then $a$ and $b$ will satisfy the relation

Smart Achievers · Accepted Answer

(1)

$∵$ ${(a x^{2} + \frac{1}{b x})}^{13}$

$= {}^{13}C_{r} {(a x^{2})}^{13 - r} {(\frac{1}{b x})}^{r} = {}^{13}C_{r} \frac{a^{13 - r}}{b^{r}} x^{26 - 3 r}$

$For coefficient of x^{8}, we have 26 - 3 r = 8 \Rightarrow r = 6$

$\Rightarrow Coefficient of x^{8} = {}^{13}C_{6} \frac{a^{7}}{b^{6}}$

$∵$ $T_{r + 1} in {(a x - \frac{1}{b x^{2}})}^{13} = {}^{13}C_{r} {(a x)}^{13 - r} {(- \frac{1}{b x^{2}})}^{r}$

$= {}^{13}C_{r} \frac{a^{13 - r}}{b^{r}} x^{13 - 3 r} {(- 1)}^{r}$

$For coefficient of x^{- 8}, we have 13 - 3 r = - 8 \Rightarrow r = 7$

$∴$ $Coefficient of x^{- 8} = - {}^{13}C_{7} \frac{a^{6}}{b^{7}}$

$Now, {}^{13}C_{6} \frac{a^{7}}{b^{6}} = - {}^{13}C_{7} \frac{a^{6}}{b^{7}} \Rightarrow a b = - 1 \Rightarrow a b + 1 = 0$

Solution

Q.
If the coefficient of $x^{8}$ in ${(a x^{2} + \frac{1}{b x})}^{13}$ is equal to the coefficient of $x^{- 8}$ in ${(a x - \frac{1}{b x^{2}})}^{13},$ then $a$ and $b$ will satisfy the relation

1 $a b + 1 = 0$

2 $a b = 1$

3 $a = 1 - b$

4 $a + b = - 1$

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Solution

Q. If the coefficient of x8 in (ax2+1bx)13 is equal to the coefficient of x-8 in (ax-1bx2)13, then a and b will satisfy the relation

1 ab+1=0

2 ab=1

3 a=1-b