So, suppose we have an algebraic curve of form:
$$ x^2 +y^2 =1$$
Now, this is actually a level set for a function $F(x,y)=x^2 +y^2$.
Suppose we took its gradient, then it's gradient would be perpendicular to this level set and hence have some direction as the normal to a curve. Once that's done consider the point on a curve $(x_o,y_o)$ and take a point on tangent line $(x,y)$ , this would give us a vector having direction of the tangent line.
$$ \vec{a} =( x-x_o) \hat{i} + (y -y_o) \hat{j}$$
If we were to dot this with the gradient then it'd be zero since dot of two perpendicular vector is zero:
$$ \nabla F_{x_o,y_o} \cdot \left[ (x-x_o) \vec{i} + (y-y_o) \vec{j} \right]$$
Or,
$$ F_x( x-x_o) + F_y (y-y_o)= 0 $$
And the equation of normal can be given as:
$$ (x, y) = (x_o, y_o) + t \nabla F_{x_o,y_o}$$
Or,
$$ \frac{x-x_o}{f_x} = \frac{y-y_o}{f_y}$$
Now is my working correct and would this method always work?