Question

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

• A. $x+3y-62=0$
• B. $x-3y-11=0$
• C. $x-3y+22=0$
• D. $x+3y+26=0$

Answer 1

Answer: (D)

Let $L$ be the common normal to parabola
$y=x^2+7x+2$ and line $y=3x-3$
$\Rightarrow$ slope of tangent of $y=x^2+7x+2$ at $P=3$
${\Rightarrow \frac{dy}{dx}]}_{ForP}=3$
$\Rightarrow 2x+7=3\Rightarrow x=-2\Rightarrow y=-8$
So $P(-2,-8)$
Normal at $P:x+3y+C=0$
$\Rightarrow C=26$ (P satisfies the line)
Normal: $x+3y+26=0$