Question : If $2x+\frac{1}{4x}=1$, then the value of $x^{2}+\frac{1}{64x^{2}}$ is:
Option 1: $0$
Option 2: $1$
Option 3: $\frac{1}{4}$
Option 4: $2$
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Correct Answer: $0$
Solution :
Given: $2x+\frac{1}{4x}=1$
Dividing by 2 on both sides,
⇒ $x+\frac{1}{8x}=\frac{1}{2}$ ----------------------------------(i)
Now, $x^{2}+\frac{1}{64x^{2}}=(x+\frac{1}{8x})^2$ – $2×x×\frac{1}{8x}$
Substituting the values, we get
⇒ $x^{2}+\frac{1}{64x^{2}}=(\frac{1}{2})^2$ – $\frac{1}{4}$
⇒ $x^{2}+\frac{1}{64x^{2}}= 0$
Hence, the correct answer is $0$.
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