(doubling rule)

(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(x _{P};y_{P})* and

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 *m _{PQ}*
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(x _{P};y_{P})* 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)

- by substituting to the each square of a variable the product between the variable itself and the corresponding coordinate of the point,
- and by substituting to each variable the half of the sum between the variable itself and the corresponding coordinate of the point.