The co-ordinates of a point on the curve $y=x \log x$ at which the normal is parallel to the line $2 x-2…

The co-ordinates of a point on the curve $y=x \log x$ at which the normal is parallel to the line $2 x-2 y=3$ are
  1. $\left(-\mathrm{e}^{-2}, 2 \mathrm{e}^{-2}\right)$
  2. $\left(-\mathrm{e}^{-2},-2 \mathrm{e}^{-2}\right)$
  3. $\quad\left(\mathrm{e}^{-2}, 2 \mathrm{e}^{-2}\right)$
  4. $\left(\mathrm{e}^{-2},-2 \mathrm{e}^{-2}\right)$

Solution

$\begin{aligned} & y=x \log x \\ \therefore \quad & \frac{\mathrm{~d} y}{\mathrm{~d} x}=1+\log x ...(i)\end{aligned}$
Slope of the normal $=-\frac{1}{\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)}=\frac{-1}{1+\log x}$ Slope of the given line is 1 . Since the normal is parallel to the given line. $\begin{aligned} \therefore \quad & \frac{-1}{1+\log x}=1 \\ & \Rightarrow \log x=-2 \\ & \Rightarrow x=\mathrm{e}^{-2} \end{aligned}$
From (i), $y=-2 \mathrm{e}^{-2}$ $\therefore \quad$ Co-ordinates of the point are $\left(\mathrm{e}^{-2},-2 \mathrm{e}^{-2}\right)$.

Asked in: MHT CET 2024 (02 May Shift 2)

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