(notes by Roberto Bigoni)
A conic γ with eccentricity e and parameter p, in the cartesian orthogonal reference system Fxy with origin in its focus F and with abscissa axis coincident with its symmetry axis and oriented outward with respect to its directrix, is described by the equation
equivalent to
If the points P(xP;yP) and Q(xQ;yQ) belong to γ, from the (1) we have
By subtracting term by term the first equation from the second one, we have
The slope of the straight line PQ is
that is
If we substitute this expression in the (3) we have
and, by simplifying,
from which
The slope of the tangent to γ at P is the limit of mPQ when Q→P:
Therefore the equation of the tangent to γ at P is given by the equation of the pencil of straight lines through P setting the slope to (6)
By expanding and rearranging the (7) we obtain
From the first one of the (2) we have also
therefore the equation of the tangent to γ at P(xP;yP) is
If we compare the equation (8) of the tangent at P with the equation (1.1) of the conic, we see that the (8) can be directly deduced from the (1.1)