Sorry for the bad title , please edit it to something better if you can.
I need a procedure to convert the general equation of ellipse - $$Ax^2 + By^2 + 2hxy + 2gx + 2fy + c = 0$$ into $$\frac{\left \{ \frac{ux+vy+q}{\sqrt{u^{2}+v^{2}}} \right \}}{a^{2}}^{2} + \frac{\left \{ \frac{vx-uy+w}{\sqrt{u^{2}+v^{2}}} \right \}}{b^{2}}^{2}=1$$ which is like $$\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$$
(Without rotating or translating . I know about rotating and translating to convert it into the above standard form (last equation) , that's not what I want).