Question 1010151
{{{x^2-2xy+y^2=2x+1}}}
Differentiate implicitly.
{{{2xdx-(2xdy+2ydx)+2ydy=2dx}}}
{{{(-2x+2y)dy=(2y-2x+2)dx}}}
{{{dy/dx=(2y-2x+2)/(2y-2x)}}}
{{{dy/dx=(y-x+1)/(y-x)}}}
So at (4,1),
{{{m=dy/dx=(1-4+1)/(1-4)=(-2)/(-3)}}}
{{{m=2/3}}}
So then,
{{{y-1=(2/3)(x-4)}}}
{{{y-1=(2/3)x-8/3}}}
{{{y=(2/3)x-8/3+3/3}}}
{{{highlight(y=(2/3)x-5/3)}}}
*[illustration dc3.JPG].