Question

If the coefficient of \(3^{\text{rd}}\) term from the beginning in the expansion of \( \left(ax^2 - \frac{8}{bx}\right)^9 \) is equal to the coefficient of \(3^{\text{rd}}\) term from the end in the expansion of \( \left(ax - \frac{2}{bx^2}\right)^9 \) then the relation between \( a \) and \( b \) is

• A. \( ab = -1 \)
• B. \( ab = 1 \)
• C. \( a^5 b^5 = -2 \)
• D. \( a^5 b^5 = 2 \)

Hint:

To find the \( k^{\text{th}} \) term from the end of \( (x+y)^n \), simply find the \( k^{\text{th}} \) term from the beginning of \( (y+x)^n \). This avoids calculating the index \( n-k+2 \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:

The \( (r+1)^{\text{th}} \) term in the expansion of \( (A+B)^n \) is given by \( T_{r+1} = \binom{n}{r} A^{n-r} B^r \). The \( k^{\text{th}} \) term from the end in the expansion of \( (X+Y)^n \) is the same as the \( k^{\text{th}} \) term from the beginning in the expansion of \( (Y+X)^n \).
Step 2: Key Formula or Approach:

For \( n=9 \), we need: 1. Coefficient of \( T_3 \) (beginning) in \( \left(ax^2 - \frac{8}{bx}\right)^9 \). 2. Coefficient of \( T_3 \) (end) in \( \left(ax - \frac{2}{bx^2}\right)^9 \), which corresponds to \( T_3 \) (beginning) of \( \left(-\frac{2}{bx^2} + ax\right)^9 \).
Step 3: Detailed Explanation:

Expansion 1: \( \left(ax^2 - \frac{8}{bx}\right)^9 \) For \( T_3 \), \( r = 2 \). \[ T_3 = \binom{9}{2} (ax^2)^{9-2} \left(-\frac{8}{bx}\right)^2 \] \[ = \binom{9}{2} (ax^2)^7 \left(\frac{64}{b^2 x^2}\right) \] \[ \text{Coefficient}_1 = \binom{9}{2} a^7 \frac{64}{b^2} \] Expansion 2: \( \left(ax - \frac{2}{bx^2}\right)^9 \) The 3rd term from the end is equivalent to the 3rd term from the beginning of \( \left(-\frac{2}{bx^2} + ax\right)^9 \). For \( T_3 \), \( r = 2 \). \[ T_3' = \binom{9}{2} \left(-\frac{2}{bx^2}\right)^{9-2} (ax)^2 \] \[ = \binom{9}{2} \left(-\frac{2}{bx^2}\right)^7 (ax)^2 \] \[ = \binom{9}{2} \left(\frac{-128}{b^7 x^{14}}\right) (a^2 x^2) \] \[ \text{Coefficient}_2 = \binom{9}{2} \left(\frac{-128}{b^7}\right) a^2 \] Equating Coefficients: \[ \binom{9}{2} a^7 \frac{64}{b^2} = \binom{9}{2} a^2 \left(\frac{-128}{b^7}\right) \] Cancel \( \binom{9}{2} \) and simplify: \[ 64 \frac{a^7}{b^2} = -128 \frac{a^2}{b^7} \] Divide both sides by \( 64 a^2 \): \[ \frac{a^5}{b^2} = \frac{-2}{b^7} \] Multiply by \( b^7 \): \[ a^5 b^5 = -2 \]
Step 4: Final Answer:

The relation is \( a^5 b^5 = -2 \).