Q52 / 170 · Manipal (MET) 2021 · Mathematics

If $9^{-x}=\frac{1}{27 \sqrt{27}}=(81)^{-y}$, then $(x, y)$ is equal to

  • Correct answer
    A
    $\left(\frac{9}{4}, \frac{9}{8}\right)$
  • B
    $\left(\frac{3}{2}, \frac{3}{4}\right)$
  • C
    $(3,6)$
  • D
    $(6,3)$

Solution

Given, $9^{-x}=\frac{1}{27 \sqrt{27}}=(81)^{-y}$ $\begin{array}{ll}\Rightarrow & (3)^{-2 x}=(3)^{-3}(3)^{-\frac{3}{2}}=(3)^{-4 y} \\ \Rightarrow & -2 x=-\frac{9}{2} \text { and }-4 y=-\frac{9}{2} \\ \Rightarrow & x=\frac{9}{4} \text { and } y=\frac{9}{8}\end{array}$

Hence, the correct answer is option: 'A'.

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