$$ (x \,\cos \alpha + y \, \sin \alpha - p)^2= (x - x_f)^2 + (y - y_f)^2 $$
Can we get a geometrical interpretation of how pedal length of straight line and focus circle center $ (x_f,y_f)$ and circle radius are related, resulting in the above parabola locus?
EDIT1:
$$ e^2 (x \,\cos \alpha + y \, \sin \alpha - p)^2= (x - x_f)^2 + (y - y_f)^2 $$
can be for other conics directly by definition.
EDIT 2:
Main motivation is to get equation of the parabola/ conic with known tilt angle (axis inclined ) to include $xy$ term.
This part X=xcosα+ysinα is rotation by angle α, so you always measure distance along a new "axis" X (if α=90 then X=x), and the RHS is just distance from a point.