Find the equation of a normal to the curve `y = x log_e x` which is parallel to the line `2x - 2y + 3 = 0.`

Options

A. (0,0)
B. `(e,e)`
C. `(e^(2),2e^(2))`
D. `(e^(-2),-2d^(-2))`

Answer

D

Explanation

Given `y=x log x implies (dy)/(dx)=1+log x`
The slope of the normal `=-1/((dy//dx))=(-1)/(1+logx)`
The slope of the line `2x-2y=3` is 1.
`:.(-1)/(1+logx)=1`
`implieslogx=-2`
`implies x=e^(-2)`, then `y=-2e^(-2)`
So, the coordinate of the point `(e^(-2),-2e^(-2))`.