Point of Contact of Confocal Conic Tangent to a Family of Parallel Lines

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To construct the conic with given foci tangent to a given line, we reflect one focus in the line, then join the reflection to the other focus. The intersection of this line with the given line is the point of contact between the conic and the tangent.

For a family of parallel lines AX+BY+C (varying C), we see the locus of the point of tangency is a hyperbola.