Question

If $g\left(x\right)=\left\{\begin{array}{l}2e\mathrm{if}\mathrm{x}\le 1\\ \log \left(x-1\right),\mathrm{if}x>1\end{array}\right.$, then the equation of the normal to $y=g\left(x\right)$ at the point $\left(3,\log 2\right)$ is

• A. $y-2x=6+\log 2$
• B. $y+2x=6+\log 2$
• C. $y+2x=6-\log 2$
• D. $y+2x=-6+\log 2$
• E. $y-2x=-6+\log 2$

Answer 1

The correct option is B

$y+2x=6+\log 2$

Explanation for the correct option:

Finding the equation of normal:

Given that,

$y=g\left(x\right)$

Since, $x=3$ which is greater than $0$, so take

$g\left(x\right)=\log \left(x-1\right)$

For slope,

Differentiate the above function with respect to $x$.

$$\begin{gathered} \frac{dy}{dx}=\frac{1}{x-1} \\ {\left(\frac{dy}{dx}\right)}_{\left(3,\log 2\right)}=\frac{1}{2} \end{gathered}$$

Slope of normal $m_2=\frac{-1}{{\displaystyle \frac{dy}{dx}}}$

$=-2$.

Now, the equation of normal at point $\left(3,\log 2\right)$,

$$\begin{gathered} y-\log 2=-2\left(x-3\right)\left[\because y-y_1=m_2\left(x-x_1\right)\right] \\ \Rightarrow y-\log 2=-2x+6 \\ \Rightarrow y+2x=6+\log 2 \end{gathered}$$

Hence, the correct option is B.