## Polar formula MP^T

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• Dear all, Can anyone tell me why polar of point P with respect to conic M is MP^T? Regards, Sung Hyun [Non-text portions of this message have been removed]
Message 1 of 2 , Mar 2, 2011
Dear all,

Can anyone tell me why polar of point P with respect to conic M is MP^T?

Regards,
Sung Hyun

[Non-text portions of this message have been removed]
• Dear Sung Hyun, Let P = (p q r) as row matrix in barycentrics and a line passing through P meets the conic at the points P1 = (p1 q1 r1) and P2 = (p2 q2 r2).
Message 2 of 2 , Mar 2, 2011
Dear Sung Hyun,
Let P = (p q r) as row matrix in barycentrics
and a line passing through P meets the conic at the
points P1 = (p1 q1 r1) and P2 = (p2 q2 r2).
The tangent at P1 is the line (x y z)MP1^t = 0
The tangent at P2 is the line (x y z)MP2^t = 0
and let Q = (p' q' r') be their intersection point.
We must have
(p' q' r')MP1^t = 0 and (p' q' r')MP2^t = 0.
The line (p' q' r')MR^t = 0 where R = (x y z)
is satisfied by P1, P2 and hence must be satisfied
by P also.This means that (p' q' r')MP^t = 0
and since the locus of Q is the polar of P
this locus must be the line (x y z)MP^t = 0
which means that the coefficients of the polar are MP^t.
Best regards