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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 to say, the curve which it always touches.

We e 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 cos2 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 cos2a+y sina cosa + m = 0, x cos2 ß+ y sin ß cosß+m=0, we find for the line joining their intersection to the focus,

x sin (a+B) - y cos (a + B) = 0.

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' = 28; 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 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

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

y'y =

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

P

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 described through PRQ must pass through F, since

the angle contained

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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

radius

=x2+m=VM+m=FM+2m=p cos 0+2m.

P

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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

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and is, therefore, one of a class of equations

p" cos no = a",

some of whose properties we shall mention hereafter.

CHAPTER XIII.

EXAMPLES AND MISCELLANEOUS PROPERTIES OF CONIC SECTIONS.

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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.

Ex. 2. Two equal rulers AB, BC, are connected by

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

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 coordinates, the centre O being the pole, and the angles being measured from the perpendicular on the fixed line; let the coordinates of A, B, be P1 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

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

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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) λ, μ, v are functions of the first degree in a, ß. But (Art. 151) the condition that Xx + μy +v should touch B is of the second degree in λ, μ, 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. x2 y2 Ans. + = 2. a2 b2

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 + =

уг

62

(~2+1=1),
1), is found from the equations mæ = a2 and ny = b2 (Art. 169).

EE

x y m n

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