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

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)

• 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.