I found the solution to one of the interesting problems of conic sections a little while ago and then I searched through this site and found an analytical approach, I am now going to provide an engineering approach and I hope to get proof in the comments
We know that for five points in the plane, four of which do not lie on the same line, there is a single conic segment passing through these points, so there is an infinite number of conic sections that Passing through four points, the centers of this family of conic sections make a one-dimensional geometric shop that turns out to be a conical section. What I did was create five points of this conic cutting using a ruler and compass.
We are looking for the geometric locale of the center of the conical section that passes through the points: $A,B,C,D$ The desired geometric locus is the conical section that passes through the following five points:
$P=AB∩CD$
$Q=AD∩BC$
$K=AC∩BD$
$M$ is mid point $B,D$
$N$ is mid point $A,C$