determine the slope of a point on a ellipse

Question

the equation of ellipse is

$Ax^2 + By^2 + Cx + Dy + Exy + F = 0$

for slope,

$2Ax+2By*dy/dx+C+D*dy/dx+Ex*dy/dx+Ey=0$

so, $(2By+D+Ex)*dy/dx=-(2Ax+C+Ey)$

=> $dy/dx=-(2Ax+C+Ey)/(2By+D+Ex)$

This should be the equation of slope at any point on a ellipse. But I found different equation from the following link

The values of dx and dy are calculated from:

dx = 2Ax + C + Ey

dy = 2By + D + Ex

The gradient/slope is calculated from:

     dy   2By + D + Ex
M  = -- = ------------
     dx   2Ax + C + Ey

(http://web.archive.org/web/20110913163032/http://web.archive.org/web/20041013054628/http://www.j3d.org/matrix_faq/curvfaq_latest.html#Q5)

My question is which one is correct?

Answer 1

Your result is correct.

In the reference the slope of normal has been given by mistake.

Take the simplest case of a circle setting all else to zero except $A,B$.

By differentiating wrt x: $ A x + B y y^ \prime = 0, $ that you gave.