The equation of the normal to the curve

Question:
The equation of the normal to the curve $y=x(2-x)$ at the point $(2,0)$ is

A. $x-2 y-2$

B. $x-2 y+2=0$

C. $2 x+y=4$

D. $2 x+y-4=0$

Solution:

Given that $y=x(2-x)$

$\Rightarrow y=2 x-x^{2}$

Slope of the tangent $\frac{d y}{d x}=2-2 x$

Slope at $(2,0)=2-4=-2$

Equation of normal:

$\left(y-y_{1}\right)=\frac{-1}{\text { Slope of tangent }}\left(x-x_{1}\right)$

$\Rightarrow(y-0)=\frac{-1}{-2}(x-2)$

$\Rightarrow 2 y=x-2$

$\Rightarrow x-2 y-2=0$

Hence option $A$ is correct.