Find the point $(x,y)$

The slope of the line $x-y=5$ can be translated with varying slopes. The point $(0,1)$ is closest to the graph at $y = e^{x}$

Find the point (x,y) on $y = e^x$ s.t. .....

Among the 4 options, i only got (0,1) is on the curve. :)

The distance of the point $(h, e^h)$ on the curve $y=e^x$ from the line $x-y=5$ is

$|\frac{h-e^h-5}{\sqrt 2 }|$

This distance will be minimum when $e^h-1=0\implies h=0,e^h=1$ . So the required point is $\boxed {(0,1)}$ .