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

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

1. (1) $x+3 y+26=0$

2. (2) $x+3 y-62=0$

3. (3) $x-3 y-11=0$

4. (4) $x-3 y+22=0$

Correct Option: 1

Solution:

The given curve is, $y=x^{2}+7 x+2$

$\Rightarrow \frac{d y}{d x}=2 x+7$

$\left(\frac{d y}{d x}\right)_{(h, k)}=2 h+7$

The tangent at $P(h, k)$ will be parallel to given line

$2 h+7=3 \Rightarrow h=-2$

Point $P(h, k)$ lies on curve

$k=(-2)^{2}-7 \times 2+2=-8$

Slope of normal at point $P(-2,-8)=-\frac{1}{3}$

$\therefore$ The equation of normal to the cuve at $P$ is

$x+3 y+26=0$