I'm asked to find the equation of the normal line for this implicitly given curve
Find an equation of the normal line to the curve
$$ \ln(2-y)=\arctan(xy)-\tan(3x)$$
at the point where $x=0$.
I can do the question fine if it was the equation of a tangent line, but how would I do the normal line?
All I can really do is given that $x=0$, I can get that $y=1$. I can then get the slope of the tangent line at $(0,1)$ which is $y'=2$. Then because the slope of the normal line would be the negative reciprocal, I get that $y'_\perp=\frac{-1}{2}$.
However, to find the equation of the normal line, I need a point on that normal line, and I have no idea how to get that. Any advice or suggestion would be useful!
Best Answer
The normal line when $x=0$ is nothing but the line with slope $-\frac 1 2$ passing through $(0,1)$ which is $y=-\frac 1 2 x+1$.