Consider the curve A*C x^2 - C*B+ A*D xy+ B*D y^2 = 0.
Compute the slope of the tangent line at the curve in P = (x_0,y_0)= (X,Y) .
y_0' (x_0)=
With F(x,y)= A*C x^2 - C*B + A*D xy+ B*D y^2 = 0 and partial derivatives
F_x und F_x one gets
\displaystyle y_0' (x_0)= - \frac{F_x(x_0,y_0)}{F_y(x_0,y_0)}.
Compute \displaystyle F_x(x,y)= 2*A*C x - C*B+ A*D y and
F_y(x,y)= -C*B- A*D x+ 2 *B*D y.
Putting in yields
\displaystyle y_0' (x_0)= - \frac{F_x(X,Y)}{F_y(X,Y)}
= fractionReduce(-2*A*C*X + (C*B+ A*D)*Y, (-C*B- A*D)*X+ 2 *B*D*Y).