

We look at the locus of the projection of the focus onto the normal, which turns about to be another parabola.



We find the coordinates of the intersection of the normals to the parabola.



We set the fixed point at the origin and the given line to be Y=b. The locus of the center of the circle is a parabola



Set the short sides along the axes with lengths a and ka. The equation is clearly a conic, and in fact a parabola:



Here is the evolute, expressed in terms of the parameter (the parameter p, being four times the distance between vertex and focus).



A is at the origin, B on the line X=a. We take the envelope of the line through B at angle η to AB. The equation is complicated, but clearly a ...



The focal chord of curvature is the chord of the circle of curvature which passes through the focus. Its length is simply expressed in terms of t.



Here is the evolute, expressed in terms of the parameter (the parameter p, being four times the distance between vertex and focus).



