Finding the equation of the normal to parabole

Question

I'm trying to solve the following problem:

Find the equation of the normal line to the parabola $y=x^2+4x+1$, which is perpendicular to the line connecting the start of coordinate system with the top of the parabola.

Can someone help me to start? I don't know how to start.

Thanks

Answer 1

As @user approach:

At first you are true on $(-2,-3)$, the vertex of parabola, Now we have: $$m_{tangent}=\frac{-3}{-2} = y_x' = (x^2+4x+1)' = 2x+4 \Longrightarrow x=-\frac{5}{4}$$ $$-\frac{2}{3}=-m_{tangent}^{-1}=m_{normal}=\frac{y-y_0}{x-x_0}=\frac{y+39/16}{x+5/4} \Longrightarrow 32x+48y=-157$$

Answer 2

HINT

We can proceed as follows

• find the vertex of the parabola, that is $(x_0,y_0)$
• determine the slope $m$ for the line connecting the origin with the vertex
• find the tangent line to the parabola with slope $m$
• determine the perpendicular to this tangent at the given point