Question

The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is

(a) x − 2y = 2
(b) x − 2y + 2 = 0
(c) 2x + y = 4
(d) 2x + y − 4 = 0

Answer 1

(a) x − 2y = 2

$$\begin{gathered} \mathrm{Here}, \\ y=x\left(2-x\right)=2x-x^2 \\ \Rightarrow \frac{dy}{dx}=2-2x \\ \text{Slope of the tangent=}{\left(\frac{dy}{dx}\right)}_{\left(2,0\right)}=2-4=-2 \\ \mathrm{S}\text{lope of the normal,}\mathit{\text{m}}\text{=}\frac{-1}{-2}\text{=}\frac{1}{2} \\ \mathrm{Given}: \\ \left(x_1,y_1\right)=\left(2,0\right) \\ \therefore \text{Equation of the normal} \\ =y-y_1=m\left(x-x_1\right) \\ \Rightarrow y-0=\frac{1}{2}\left(x-2\right) \\ \Rightarrow 2y=x-2 \\ \Rightarrow x-2y=2 \end{gathered}$$