# If the normal to the curve y=xlnx is parrallel to the straight line 2x-2y+3=0 ,then the normal equation is...... a.x-y=3e^-2 b.x-y=6e^-2 c.x-y=3e^2 ?

May 10, 2018

Option A is correct .

#### Explanation:

The line can be rewritten a

$2 x + 3 = 2 y$

$x + \frac{3}{2} = y$

So the slope of the line (as well as the normal line) will be $1$. This means the slope of the tangent will be $- 1$, so we have to set the derivative to $- 1$.

$y ' = \ln x + x \left(\frac{1}{x}\right) = \ln x + 1$

We have:

$- 1 = \ln x + 1 \to \ln x = - 2 \to x = {e}^{-} 2$

The corresponding value of $y$ is $y = {e}^{-} 2 \ln \left({e}^{-} 2\right) = - 2 {e}^{-} 2$

We now can see that the normal has equation

$y - \left(- 2 {e}^{-} 2\right) = x - {e}^{-} 2$

$y = x - {e}^{-} 2 - 2 {e}^{-} 2$

$y = x - 3 {e}^{-} 2$

Or

$x - y = 3 {e}^{-} 2$

Which is option A.

Hopefully this helps!