Tangent to a conic at one of its points
(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

Eqn001.gif

equivalent to

Eqn015.gif

If the points P(xP;yP) and Q(xQ;yQ) belong to γ, from the (1) we have

Eqn002.gif

By subtracting term by term the first equation from the second one, we have

Eqn003.gif

Eqn004.gif

The slope of the straight line PQ is

Eqn005.gif

that is

Eqn006.gif

If we substitute this expression in the (3) we have

Eqn007.gif

and, by simplifying,

Eqn008.gif

from which

Eqn009.gif

The slope of the tangent to γ at P is the limit of mPQ when Q→P:

Eqn010.gif

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)

Eqn011.gif

By expanding and rearranging the (7) we obtain

Eqn012.gif

From the first one of the (2) we have also

Eqn013.gif

therefore the equation of the tangent to γ at P(xP;yP) is

Eqn014.gif

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)