Deriving the Polar of a Point on a Conic

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Discussion Overview

The discussion revolves around the derivation of the polar of a point with respect to a general conic. It includes theoretical aspects of conics, definitions of the polar, and the implications of a point lying on the conic.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant presents the equation of a general conic and the formula for the polar of a point on that conic.
  • Another participant questions the definition of the polar of a point and its behavior when the point lies on the conic.
  • A participant acknowledges a mistake regarding the point's position on the conic and clarifies that the polar represents the chord of contact of tangents from the point.
  • There is a query about the equation of the tangent line from the point to the conic.
  • One participant expresses that they are beginning to understand the derivation, mentioning challenges with implicit differentiation and other oversights.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the derivation process, and multiple viewpoints regarding the definition and implications of the polar remain present.

Contextual Notes

There are unresolved assumptions regarding the conditions under which the polar is derived, particularly concerning the position of the point relative to the conic.

Appleton
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If a general conic is

<br /> ax^2+2hxy+by^2+2gx+2fy+c=0<br />

I am told that, if P(p, q) is a point on this conic, then the polar of P(p, q) to this conic is

<br /> apx+h(py+qx)+bgy+g(p+x)+f(q+y)+c=0<br />

How is this derived?
 
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What is your definition of the polar of a point wrt the conic? What do you know about the polar if the point lies on the conic?
 
Thanks for your reply micromass. I realize I made a mistake. P(p,q) does not lie on the conic.

The polar is the chord of contact of the tangents from P.

If the point lies on the conic then the chord of contact would be non existent as P and the tangent points would all be coincident.

If we assume there are no constraints on P, what would be the derivation?
 
Do you know the equation of the tangent line from ##P##?
 
OK I think I'm with you now. Thanks for the prompt. I think the implicit derivative was my main stumbling block, amongst various other oversights.
 

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