[alogo] A property of the directrix

Every line through a point C of the directrix of a conic intersects the conic at two points {D,E} such that angle(DAE) is bisected by the polar pC of C.
Here A is the focal point corresponding to the directrix i.e. the pole of this directrix.

[0_0] [0_1]
[1_0] [1_1]
[2_0] [2_1]
[3_0] [3_1]

This is a consequence of the characteristic property of the directrix to have the ratios DXD/DA = e, constant.
Here XD is the projection of D on the directrix and e is the eccentricity of the conic.
Let F be the intersection of DE with the polar pC of C. Then by the characteristic property of polars FD/FE = - CD/CE = -DXD/EXE = - DA/EA.
This means that F is on the bisector of angle(DAE).
Corollary Lines AC and AF are orthogonal.
Application To construct a conic (c) knowing two tangents of it, its contact point with one of them and one of its foci. Look at ConicConstruction.html for the solution.

See Also

ConicConstruction.html

Return to Gallery


Produced with EucliDraw©