Question

Find the equations of tangent and normal to the curves at the indicated points on it.
$y=x^2+4x+1$ at $(-1,-2)$

Answer 1

Given equation curve

$y=x^2+4x+1$

$\Rightarrow \frac{dy}{dx}=2x+4$

${\left(\frac{dy}{dx}\right)}_{-1,-2}=2x-1+4$

i.e., equation of tangent is

$y=mx+C$

$m=\frac{dy}{dx}$

$\Rightarrow y=2x+C$

at $(-1,-2)C$ is

$-2=-2+C$

$C=0$

$\Rightarrow y=2x$ is tangent to curve at $(-1,-2)$

$mxm=-1$

$\Rightarrow m=\frac{-1}{m}=-1/2$

$y=mx+c$ is equation of normal

$y=1/2x+C$

$\therefore C$ at $(-1,-2)$ is

$-2=1/2+C\Rightarrow C=-5/2$

$\therefore x+2y+5=0$ is equation of normal.