Question

Let $f\left(x\right)=lnmx(m>0)$ and $g\left(x\right)=px$. Then the equation $|f\left(x\right)|=g\left(x\right)$ has exactly two solutions (not necessarily distinct) for

• A. $p=\frac{m}{e}$
• B. $p=\frac{e}{m}$
• C. $0<p\le \frac{e}{m}$
• D. $0<p\le \frac{m}{e}$

Answer 1

The correct option is A $p=\frac{m}{e}$

from the figure:
$|f\left(x\right)|=g\left(x\right)$ will have exactly two solutions when $g\left(x\right)=px$ is tangent to $f\left(x\right)=\ln mx$
let $P(h,k)$ be a point on $f\left(x\right)=\ln mx$
then, $k=\ln mh$
$\frac{dy}{dx}$ at point P $=\frac{1}{h}$
Equation of tangent at point P $y-k=\frac{(x-h)}{h}$
Since, it passes through origin
Therefore, $k=1$ & $h=e/m$
and $y=x/h$ $\Rightarrow y=mx/e$
on comparing with $g\left(x\right)=px$
we get, $p=m/e$

Ans: A