The polar of some point P wrt some conic CO is the line L connecting the points of tangency of the tangents through P at CO.
In this construction P is called the pole and L is called the polar.
This construction is very intuitive. However there is a flaw in the definition because the polar cannot be constructed under all circumstances. For example when CO is some ellipse and P is drawn within the ellipse, then the construction fails.
Therefore another construction is made that includes the result of the first construction:
The polar of some point P wrt some conic is the 3rd diagonal (see QG-L1 in EQF) of any quadrigon formed by the intersection points of any 2 lines through P with the conic.
In this construction P is called the pole and L is called the polar.
This construction is very intuitive. However there is a flaw in the definition because the polar cannot be constructed under all circumstances. For example when CO is some ellipse and P is drawn within the ellipse, then the construction fails.
Therefore another construction is made that includes the result of the first construction:
The polar of some point P wrt some conic is the 3rd diagonal (see QG-L1 in EQF) of any quadrigon formed by the intersection points of any 2 lines through P with the conic.
Properties:
• The polar line of the focus of a parabola is the directrix
• The polar line of the focus of a parabola is the directrix