Deriving the Polar of a Point on a Conic

[Appleton]

If a general conic is

$ax^2+2hxy+by^2+2gx+2fy+c=0$

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

$apx+h(py+qx)+bgy+g(p+x)+f(q+y)+c=0$

How is this derived?

 

[micromass]

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?

 

[Appleton]

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?

 

[micromass]

Do you know the equation of the tangent line from $P$?

 

[Appleton]

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.