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.
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