Question

The minimum distance between a point on the curve $y=e^x$ and a point on the curve $y={\log }_{e}x$ is

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

Answer 1

Answer: (B)

Since $y=e^x$ and $y={\log }_{e}x$ are inverse to each other their graphs are symmetrical about $y=x$.
Minimum distance between the curves is the distance between the points on the curves where tangent is parallel to $y=x$ or slope of tangent is $1$ .
Now for $y=e^x,y^{\prime }=e^x$, for $e^x=1,x=0$, so the point on the curve $y=e^x$ is $(0,1)$ and symmetric point on the curve $y={\log }_{ex}$ is $(1,0)$
Distance between these points is $\sqrt{2}$.