Question

The point on the curve $y=e^x$ which is at shortest distance from the line $y=x-4,$ is

• A. $(0,0)$
• B. $(1,e)$
• C. $(0,1)$
• D. $(-1,\frac{1}{e})$

Answer 1

The correct option is C $(0,1)$

For shortest distance between curve and line, the slope of tangent at curve should be equal to slope of line.
$\therefore {\left(\frac{dy}{dx}\right)}_{c_1}={\left(\frac{dy}{dx}\right)}_{c_2}$
$\Rightarrow e^x=1$
$\Rightarrow x=0,$ putting in $y=e^x$
$\Rightarrow y=1$
So, Point $(0,1)$ is at the shortest distance.