Given x+xy+2y^2=6 find the coordinates of all other points on this curve with slope equal to -1/3. Please show work.

Answer 1

Given 2y^2+xy+x=6 -------(1). Differentiate both sides to obtain 4ydy/dx+xdy/dx+y+1 = 0 or dy/dx = -(y+1)/(x+4y) = -1/3. This gives us x+4y-3(y+1)=0 -----------(2). Solve equations (1) & (2) simultaneously to obtain the two points as (2,1)&(6,-3)