Ellipse

From a problem by JC Arbaut on fr.sci.maths
Given a line (AB), the circle C with diameter [AB], centered in O, midpoint of [AB].
Let the lines D1 and D2, perpendicular to (AB) respectively through A and B.
Let any point M on circle C.

Draw AM intersecting D2 in P. Draw BM intersecting D1 in Q.
Draw the perpendicular to (AB) through M, intersecting (PQ) in N.
Then, when M dscribes circle C, point N describes an ellipse, and PQ is tangent in N to this ellipse.
Properties of this ellipse.
Hint

Draw line (OM), intersecting D1 in R (except for two locations of M on the circle). Then BR is perpendicular to PQ.

Extension

Given any two lines SA and SB, and any circle (C).
Let M any point on the circle. AM intersects SB in P, BM intersects SA in Q, SM intersects PQ in N
N describes a conic section when M describes the circle.
Condition for this conix section to be an ellipse, a parabola, an hyperbola ?
Condition for PQ to be tangent in N to this conic section ?

Details

 

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