is parallel to the parabola's axis, hence intersects the parabola at one point only.

For a secant BLK of the parabola parallel to a fixed direction e from an arbitrary point B the following relation holds

and is independent of the position of B, depending only on the direction of e:

(BK*BL)/BA' = k (constant).

Here A' is the unique intersection point of the parabola with the line parallel to the axis of the

parabola and passing through B (the diameter of the parabola through B).

To prove it represent the parabola in coordinates and in the form:

y = ax

with a constant a. Then apply first the well known relation (see PowerGeneral.html ) for the two secants

KBL, C'BD' by which the ratio r

direction e of line KBL. Then calculate the ratio r

by the symmetry x

-(x

Note that, by the aforementioned reference, the ratio r

determines the direction of the secant. Hence the ratio in total is r = (sin(fi)-a*cos

A similar situation occuring in hyperbolas is discussed in PowerGeneralHyperbola.html . [SalmonConics, p. 151]

passing through {B,C}, having axis parallel to the parallel sides and having AD as its tangent at P.

Locate the position P=L on AD for which such a construction is possible.

Assume L found and project {B,C} on the parallel e to AB from L. Then take the symmetrics {B',C'} of {B,C}

with respect to {N,M} respectively. The parabola has to pass through points {B,C',L,C,B} hence can be

constructed as a conic passing through five points.

By the previous section the ratios s

equal, thus implying that:

LD

This is then a necessary condition and is satisfied by exactly one point L on AD.

The condition is also sufficient, since setting the value of the common ratio s

LD

We find easily that the five points {B,C',L,C,B'} are on the parabola, which in skew axes coinciding

with LD (x-axis) and LM (y-axis) is represented by ( see ParabolaSkew.html ):

x

of P on line AD we obtain conics passing through {P,B,C,B',C'}, tangent to AD at P and having the line

e

By applying the results obtained in PowerGeneral.html we locate the other intersection point Q of the

conic with line e

The case of the parabola is an exceptional one for P=L and for which point Q is at infinity.

Another exceptional case occurs for P=E', where E' is the harmonic conjugate of E with respect to {A,D}.

In this case Q is coincident with P. For all other cases we obtain conics, which depending on the position

of P, relative to L are ellipses (as shown) or hyperbolas. The two cases of conics are separated by the

location of L. For points on the half-line LE, containing the harmonic E' of E we obtain hyperbolas.

For points on the half-line LA, not containing E', we obtain ellipses.

PowerGeneralHyperbola.html

see ParabolaSkew.html

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