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

=0;

x cos2a + y sina cosa + m = 0, x cos2 8+ y sin ẞ cosß+m=

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=2a and VFP' = 2ß; therefore the line making an angle with the axis = a+ẞ 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 co-ordinates 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 Co-ordinate 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. 178.

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.

P

R

Q

For the circle described through T

PRQ must pass

through F, since

the angle contained

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 e1 (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

and is, therefore, one of a class of equations,

p" cosno = a",

some of whose properties we shall mention hereafter.

CHAPTER XIII.

EXAMPLES AND MISCELLANEOUS PROPERTIES OF CONIC SECTIONS.

225. THE method of applying algebra to problems relating to conic sections is essentially the same as that employed in the case of the right line and circle, and will present no difficulty to any reader who has carefully worked out the Examples given in Chapters III. and VII. We, therefore, only think it necessary to select a few out of the great multitude of examples which lead to loci of the second order, and we shall then add some properties of conic sections, which it was not found convenient to insert in the preceding chapters.

Ex. 1. Through a fixed point P is drawn a line LK (see fig., p. 40) terminated by two given lines. Find the locus of a point Q taken on the line, so that PL = QK.

a pivot at B; the extremity A is fixed, while the extremity C is made to traverse the right line AC; find the locus described by any fixed point P on BC.

Ex. 3. Given base and the product of the tangents of the halves of the base angles of a triangle: find the locus of vertex.

A

B

P

Expressing the tangents of the half angles in terms of the sides, it will be found that the sum of sides is given; and, therefore, that the locus is an ellipse, of which the extremities of the base are the foci.

Ex. 4. Given base and sum of sides of a triangle; find the locus of the centre of the inscribed circle.

It may be immediately inferred, from the last example, and from Ex. 4, p. 47, that the locus is an ellipse, whose vertices are the extremities of the given base.

Ex. 5. Given base and sum of sides, find the locus of the intersection of bisectors of sides.

Ex. 6. Find the locus of the centre of a circle which makes given intercepts on two given lines.

Ex. 7. Find the locus of the centre of a circle which touches two given circles; or which touches a right line and a given circle.

Ex. 8. Find locus of centre of a circle which passes through a given point and makes a given intercept on a given line.

Ex. 9. Or which passes through a given point, and makes on a given line an intercept subtending a given angle at that point.

Ex. 10. Two vertices of a given triangle move along fixed right lines; find the locus of the third.

Ex. 11. A triangle ABC circumscribes a given circle; the angle at C is given, and B moves along a fixed line; find the locus of A.

Let us use polar co-ordinates, the centre O being the pole, and the angles being measured from the perpendicular on the fixed line; let the co-ordinates of A, B, be p, 0; p', '. Then we have p' cos 0' p. But it is easy to see that the angle AOB is given (= a). And since the perpendicular of the triangle AOB is given, we have

But 0+0= a; therefore the polar equation of the locus is

p2:

p2p2 sin2 a

p2 cos2 (a − 0) + p2 – 2pp cos a cos (a: — 0) '

which represents a conic.

Ex. 12. Find the locus of the pole with respect to one conic A of any tangent to another conic B.

Let aß be any point of the locus, and λx +μy+v its polar with respect to the conic A, then (Art. 89) X, μ, v are functions of the first degree in a, ß. But (Art. 151) the condition that Xx + y +v should touch B is of the second degree in λ, u, V. The locus is therefore a conic.

Ex. 13. Find the locus of the intersection of the perpendicular from a focus on any tangent to a central conic, with the radius vector from centre to the point of contact. Ans. The corresponding directrix.

Ex. 14. Find the locus of the intersection of the perpendicular from the centre on any tangent, with the radius vector from a focus to the point of contact. Ans. A circle.. Ex. 15. Find the locus of the intersection of tangents at the extremities of conjugate diameters. = 2.

Ans. 22+1=

a2

This is obtained at once by squaring and adding the equations of the two tangents, attending to the relations Art. 172.

Ex. 16. Trisect a given arc of a circle. The points of trisection are found as the intersection of the circle with a hyperbola. See Ex. 7, p. 47.

Ex. 17. One of the two parallel sides of a trapezium is given in magnitude and position; and the other in magnitude. The sum of the remaining two sides is given; find the locus of the intersection of diagonals.

Ex. 18. One vertex of a parallelogram circumscribing an ellipse moves along one directrix; prove that the opposite vertex moves along the other, and that the two remaining vertices are on the circle described on the axis major as diameter.

226. We give in this Article some examples on the focal properties of conics.

Ex. 1. The distance of any point on a conic from the focus is equal to the whole length of the ordinate at that point, produced to meet the tangent at the extremity of the focal ordinate.

Ex. 2. If from the focus a line be drawn making a given angle with any tangent, find the locus of the point where it meets it.

Ex. 3. To find the locus of the pole of a fixed line with regard to a series of concentric and confocal conic sections.

We know that the pole of any line (

y

n

+ =1 1), with regard to the conic

is found from the equations mx = a2 and ny = b2 (Art. 169).

Now, if the foci of the conic are given, a2 - b2= c2 is given; hence, the locus of the pole of the fixed line is

mx - ny = c2,

the equation of a right line perpendicular to the given line.

If the given line touch one of the conics, its pole will be the point of contact. Hence, given two confocal conics, if we draw any tangent to one and tangents to the second where this line meets it, these tangents will intersect on the normal to the first conic.

Ex. 4. Find the locus of the points of contact of tangents to a series of confocal ellipses from a fixed point on the axis major. Ans. A circle. Ex. 5. The lines joining each focus to the foot of the perpendicular from the other focus on any tangent, intersect on the corresponding normal and bisect it.

Ex. 6. The focus being the pole, prove that the polar equation of the chord through points whose angular co-ordinates are a + ß, a − ß, is

Ρ 2p

e cose + secẞ cos (0 — a).

This expression is due to Mr. Frost (Cambridge and Dublin Math. Journal, 1., 68, cited by Walton, Examples, p. 375). It follows easily from Ex. 3, p. 37.

Ex. 7. The focus being the pole, prove that the polar equation of the tangent, at the point whose angular co-ordinate is a, is 2 = e cose + cos (0 − a).

2p

This expression is due to Mr. Davies (Philosophical Magazine for 1842, p. 192, cited by Walton, Examples, p. 368).

Ex. 8. If a chord PP' of a conic pass through a fixed point 0, then

The reader will find an investigation of this theorem by the help of the equation of Ex. 6 (Walton's Examples, p. 377). I insert here the geometrical proof given by Mr. Mac Cullagh, to whom, I believe, the theorem is due. Imagine a point O taken anywhere on PP' (see figure, p. 195), and let the distance FO be e' times the distance of O from the directrix; then since the distances of P and O from the directrix are proportional to PD and OD, we have

or, since (Art.191) PFT is half the sum, and OFT half the difference, of PFO and P'FO,

It is obvious that the product of these tangents remains constant if O be not fixed, but be anywhere on a conic having the same focus and directrix as the given conic.

Ex. 9. To express the condition that the chord joining two points x'y', x"y" on the curve passes through a focus.

This condition may be expressed in several equivalent forms, two of the most useful of which are got by expressing that "'+ 180° where e', 0" are the angles made with the axis by the lines joining the focus to the points. The condition sine"=- - sin e' gives