Re Pencil of conics
The two conics intersect in 4 pts.
Choose any other point in the plane to give the 5th point of another conic
of the pencil.
It's a slightly different proposition if the two conics intersect in
Then you have to determine how to draw the conic.
I haven't thought this through, but I guess it involves the double points of
an elliptic involution.
I would go to Hatton's Geometry of the Imaginary for that.
It's available on the Internet Archive
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