Some chords have the same form of equations for all conics, i.e. a circle, an ellipse, a hyperbola or a parabola, when written in terms of the equations of the conics and their tangents. For example, the equation of a chord of a non-degenerate conic with a fixed midpoint $x_0,y_0$ has the equation $T(x_0, y_o)=S(x_0,y_0)$, where T and S are the equations of the tangent and the conic respectively referred to the point $x_0,y_0$t.
I can see that these chords are projectively same. Is there a way to get this equation directly (instead of finding their equations separately for a parabola, ellipse, hyperbola and showing that it is true individually) for all conics using projective geometry? I suspect that there is a way to do it using homogeneous coordinates. But I have just started learning homogeneous coordinates and am unable to do it with my present knowledge. Can anyone help?