Question : The third proportional of $a$ and $\frac{b^4}{4a}$ is:
Option 1: $\frac{b^8}{16 a^2}$
Option 2: $\frac{b^8}{8 a^2}$
Option 3: $\frac{b^8}{8 a^3}$
Option 4: $\frac{b^8}{16 a^3}$
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Correct Answer: $\frac{b^8}{16 a^3}$
Solution :
Let the third proportional of $a$ and $\frac{b^4}{4 a}$ be $x$
So, $a : \frac{b^4}{4 a} :: \frac{b^4}{4 a} : x$
⇒ $\frac{a}{\frac{b^4}{4 a}}=\frac{\frac{b^4}{4 a}}{x}$
⇒ $x = \frac{\frac{b^4}{4 a}\times \frac{b^4}{4 a}}{a}$
⇒ $x= \frac{b^8}{16 a^3}$
Hence, the correct answer is $\frac{b^8}{16 a^3}$.
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