To avoid fractions we shall often, in the following Articles, use the abbreviation m = p.

1p.

To find the distance of any point on the curve from the focus. The coordinates of the focus being (m, 0), the square of its distance from any point is

(x′ — m)2 + y22 = x22 – 2mx′ + m2 + 4mx′ = (x2 + m)2. Hence the distance of any point from the focus · x2 + m.

=

This enables us to express more simply the result of Art. 212, and to say that the parameter of any diameter is four times the distance of its extremity from the focus.

215. The polar of the focus of a parabola is called the directrix, as in the ellipse and hyperbola.

Since the distance of the focus from the vertex = m, its polar is (Art. 211) a line perpendicular to the axis at the same distance on the other side of the vertex. The distance of any point from the directrix must, therefore, =x′+m.

Hence, by the last Article, the distance of any point on the curve from the directrix is equal to its distance from the focus.

We' saw (Art. 186) that in the ellipse and hyperbola the distance from the focus is to the distance from the directrix in the constant ratio e to 1. We see, now, that this is true for the parabola also, since in the parabola e = 1 (Art. 209).

The method given for mechanically describing an hyperbola, Art. 202, can be adapted to the mechanical description of the parabola, by simply making the angle ABR a right angle.

216. The point where any tangent cuts the axis, and its point of contact, are equally distant from the focus.

For, the distance from the vertex of the point where the tangent cuts the axis = (Art. 210), its distance from the focus is therefore x' + m.

. 217. Any tangent makes equal angles with the axis and with the focal radius vector.

This is evident from inspection of the isosceles triangle, which, in the last Article, we proved was formed by the axis, the focal radius vector, and the tangent.

This is only an extension of the property of the ellipse (Art. 188), that the angle TPF=T'PF'; for, if we suppose the

focus F' to go off to infinity, the line PF" will become parallel to the axis, and TPF=PTF. (See figure, p. 200)

Hence the tangent at the extremity of the focal ordinate cuts the axis at an angle of 45°.

218. To find the length of the perpendicular from the focus on the tangent.

The perpendicular from the point (m, 0) on the tangent {yy' = 2m (x+x'}} is

It appears, also, from this expression and from Art. 213 that FR is half the normal, as we might have inferred geometrically from the fact that TF- FN.

219. To express the perpendicular from the focus in terms of the angles which it makes with the axis.

The equation of the tangent, the focus being the origin, can therefore be expressed

and hence we can express the perpendicular from any other point in terms of the angle it makes with the axis.

220. The locus of the extremity of the perpendicular from the focus on the tangent is a right line.

For, taking the focus for pole, we have at once the polar equation

which obviously represents the tangent at the vertex.

Conversely, if from any point F we draw FR a radius vector

to a right line VR, and draw PR perpendicular to it, the line FR will always touch a parabola having F for its focus.

We shall show hereafter how to solve generally questions of this class, where one condition less than is sufficient to determine a line is given, and it is required to find its envelope, that is the curve which it always touches.

to say,

We leave, as a useful exercise to the reader, the investigation of the locus of the foot of the perpendicular by ordinary rectangular coordinates.

221. To find the locus of the intersection of tangents which cut at right angles to each other.

The equation of any tangent being (Art. 219)

x cos ay sina cosa + m =

:0;

the equation of a tangent perpendicular to this (that is, whose perpendicular makes an angle = 90°+a with the axis) is found by substituting cosa for sina, and - sina for cosa, or

x sin ay sina cosa + m = 0.

a is eliminated by simply adding the equations, and we get

x+2m = 0,

the equation of the directrix, since the distance of focus from directrix = 2m.

222. The angle between any two tangents is half the angle between the focal radii vectores to their points of contact.

For, from the isosceles PFT, the angle PTF, which the tangent makes with the axis, is half the angle PFN, which the focal radius makes with it. Now, the angle between any two tangents is equal to the difference of the angles they make with the axis, and the angle between two focal radii is equal to the difference of the angles which they make with the axis.

The theorem of the last Article follows as a particular case of the present theorem: for if two tangents make with each other an angle of 90°, the focal radii must make with each other an angle of 180°, therefore the two tangents must be drawn at the extremities of a chord through the focus, and, therefore, from the definition of the directrix, must meet on the directrix.

223. The line joining the focus to the intersection of two tangents bisects the angle which their points of contact subtend at the focus.

Subtracting one from the other, the equations of two tangents, viz.

x cos ay sina cosa + m = 0, x cos2 ß + y sin ß cosẞ+m=0, we find for the line joining their intersection to the focus,

This is the equation of a line making the angle a+ẞ with the axis of x. But since a and B are the angles made with the axis by the perpendiculars on the tangent, we have VFP=22 and VFP'=28; therefore the line making an angle with the axis =a+B must bisect the angle PFP'. This theorem may also be proved by calculating, as in Art. 191, the angle (0-0) subtended at the focus by the tangent to a parabola from the point xy, when it will be found that cos (0-0) = a value which, being

=

x + m

ρ

independent of the coordinates of the point of contact, will be the same for each of the two tangents which can be drawn through xy. (See O'Brien's Coordinate Geometry, p. 156.)

Cor. 1. If we take the case where the angle PFP' = 180°, then PP' passes through the focus; the tangents TP, TP' will intersect on the directrix, and the angle TFP=90° (See Art. 192). This may also be proved directly by forming the equations of the polar of any point (-m, y) on the directrix, and also the equation of the line joining that point to the focus. These two equations are

y'y = 2m (x — m), ́2m (y−y') + y′ (x + m) = 0,

which obviously represent two right lines at right angles to each other.

COR. 2. If any chord PP' cut the directrix in D, then FD is the external bisector of the angle PFP'. This is proved as at p. 184.

Cor. 3. If any variable tan

T

D

F

P'

gent to the parabola meet two fixed tangents, the angle subtended at the focus by the portion of the variable tangent intercepted between the fixed tangents is the supplement of the angle between the fixed tangents. For (see next figure).

the angle QRT is half pFq (Art. 222), and, by the present Article, PFQ is obviously also half pFq, therefore PFQ is =QRT, or is the supplement of PRQ.

COR. 4. The circle circumscribing the triangle formed by any

three tangents to a

parabola will pass

through the focus. For the circle de

scribed through T

PRQ must pass

through F, since

the angle contained

P

R

Ο

F

in the segment PFQ will be the supplement of that contained in PRQ.

224. To find the polar equation of the parabola, the focus being the pole.

We proved (Art. 214) that the focal

P

This is exactly what the equation of Art. 193 becomes, if we suppose e=1 (Art. 209). The properties proved in the Examples to Art. 193 are equally true of the parabola.

In this equation is supposed to be measured from the side FM; if we suppose it measured from the side FV, the equation becomes

and is, therefore, one of a class of equations

p" cos no = a",

some of whose properties we shall mention hereafter.