Sequences And Series Ques 19
- If the harmonic mean and geometric mean of two positive numbers are in the ratio $4: 5$. Then, the two numbers are in the ratio… .
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Answer:
Correct Answer: 19.$(4:1)$
Solution: Let the two positive numbers be $k a$ and $a, a>0$.
Then, $\quad G=\sqrt{k a \cdot a}=\sqrt{k} \cdot a$
and $\quad H=\frac{2(k a) a}{k a+a}=\frac{2 k a}{k+1}$
Again, $\quad \frac{H}{G}=\frac{4}{5} \quad $ [given]
$\Rightarrow \quad \frac{\frac{2 k a}{k+1}}{\sqrt{k} a}=\frac{4}{5} \Rightarrow \frac{2 \sqrt{k}}{k+1}=\frac{4}{5}$
$\Rightarrow \quad 5 \sqrt{k}=2 k+2$
$\Rightarrow \quad 2 k-5 \sqrt{k}+2=0$
$\Rightarrow \quad \sqrt{k}=\frac{5 \pm \sqrt{25-16}}{4}=\frac{5 \pm 3}{4}=2, \frac{1}{2}$
$\Rightarrow \quad k=4,1 / 4$.
Hence, the required ratio is $4: 1$.
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