Question

If the curves $C_1:y=\frac{\ln x}{x}$ and $C_2:y=\lambda x^2$ (where $\lambda$ is constant) touches each other, then the value of $\lambda$ is

• A. $\frac{1}{e}$
• B. $\frac{1}{2e}$
• C. $\frac{1}{3e}$
• D. $\frac{1}{4e}$

Answer 1

Answer: (C)

Since, $C_1$ and $C_2$ touches each other.
$\Rightarrow \frac{\ln x}{x}=\lambda x^2$
$\Rightarrow \ln x=\lambda x^3$....(1)
And $\frac{1-\ln x}{x^2}=2\lambda x\Rightarrow 1-\lambda x^3=2\lambda x^3$
$\Rightarrow 3\lambda x^3=1\Rightarrow x={\left(\frac{1}{3\lambda }\right)}^{\frac{1}{3}}=(3\lambda {)}^{\frac{-1}{3}}$
From (1), $\frac{-1}{3}\ln (3\lambda )=\lambda \cdot \frac{1}{3\lambda }\Rightarrow \ln (3\lambda )=-1.\Rightarrow \lambda =\frac{1}{3e}$