Question

$x^2-xy+y^2=3$ is the equation of an ellipse. By implicit differentiation determine the equation of the normal of the given equation at (-1,1)

Answer 1

Let $F(x,y)=x^2-xy+y^2-3.$ Then $F_x=2x-y$ and $F_y=-x+2y,$ and hence $(F_x(-1,1),F_y(-1,1))=(-3,3)$ is a normal vector to the curve at the point $(-1,1).$ It follows that the equation of the normal at $(-1,1)$ is $\frac{x+1}{-3}=\frac{y-1}{3},$ which is equivalent to $y=-x.$