Let P(h,k) be a point on the curve


Let $\mathrm{P}(\mathrm{h}, \mathrm{k})$ be a point on the curve $\mathrm{y}=\mathrm{x}^{2}+7 \mathrm{x}+2$, nearest to the line, $\mathrm{y}=3 \mathrm{x}-3$. Then the equation of the normal to the curve at $P$ is :

  1. $x+3 y-62=0$

  2. $x-3 y-11=0$

  3. $x-3 y+22=0$

  4. $x+3 y+26=0$

Correct Option: , 4


Let $\mathrm{L}$ be the common normal to parabola $y=x^{2}+7 x+2$ and line $y=3 x-3$

$\Rightarrow$ slope of tangent of $y=x^{2}+7 x+2$ at $P=3$

$\left.\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}\right]_{\text {For } \mathrm{P}}=3$

$\Rightarrow 2 x+7=3 \Rightarrow x=-2 \Rightarrow y=-8$

So $\mathrm{P}(-2,-8)$

Normal at $\mathrm{P}: \mathrm{x}+3 \mathrm{y}+\mathrm{C}=0$

$\Rightarrow \mathrm{C}=26$ (P satisfies the line)

Normal: $x+3 y+26=0$

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