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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.

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

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 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 p, e; 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

p2 =

p2p2 sin2 a

p2 cos2 (a) +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) A, μ, v are functions of the first degree in a, B. But (Art. 151) the condition that Ax + μy + 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 conju gate diameters.

x2 y2 Ans. + a2 b2

= 2.

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 con. centric and confocal conic sections.

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

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Now, if the foci of the conic are given, a2 - b2 = c 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 coordinates are a + ß, a −

-B, is

P
= e cos 0+ sec ẞ cos (0 − a).
2p

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 p 2p

the point whose angular coordinate is α, is = e cos 0 + cos (0 - a).

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

is constant.

tan PFO.tan P'FO

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 0 taken anywhere on PP' (see figure p. 206), 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

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

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It is obvious that the product of these tangents remains constant if ◊ 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 the focus.

=

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

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Ex. 10. If normals be drawn at the extremities of any focal chord, a line drawn through their intersection parallel to the axis major will bisect the chord. [This solution is by Larrose, Nouvelles Annales, XIX. 85.]

Since each normal bisects the angle between the focal radii, the intersection of normals at the extremities of a focal chord is the centre of the circle inscribed in the triangle whose base is that chord, and sides the lines joining its extremities to the other focus. Now if a, b, c be the sides of a triangle whose vertices are x'y', x''y'', x'"'y'', then, Ex. 6, p. 6, the coordinates of the centre of the inscribed circle are

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In the present case the coordinates of the vertices are x', y'; x", y"; - c, 0; and the lengths of opposite sides are a + ex'", a + ex', 2a - ex' - ex". We have therefore

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or, reducing by the first relation of the last Example, y = (y' + y′′), which proves the theorem.

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We could find, similarly, expressions for the coordinates of the intersection of tangents at the extremities of a focal chord, since this point is the centre of the circle exscribed to the base of the triangle just considered. The line joining the intersection of tangents to the corresponding intersection of normals evidently passes through a focus, being the bisector of the vertical angle of the same triangle.

Ex. 11. To find the locus of the intersection of normals at the extremities of a focal chord.

Let a, B be the coordinates of the middle point of the chord, and we have, by the last Example,

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a2 (x + c)
a2 + c2

'; B = {(y' + y′′) = y.

If, then, we knew the equation of the locus described by aß, we should, by making the above substitutions, have the equation of the locus described by xy. Now the polar equation of the locus of middle point, the focus being origin, is (Art. 193)

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which, transformed to rectangular axes, the centre being origin, becomes

b2a2 + a2ß2 = b2ca.

The equation of the locus sought is, therefore,

a2b2 (x + c)2 + (a2 + c2)2y2 = b2c (a2 + c2) (x + c).

Ex. 12. If be the angle between the tangents to an ellipse from any point P, and if p, p' be the distances of that point from the

foci, prove that cos 0

p2 + p22
2pp'

4a2

T

(see also

Art. 194 d).

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Ex. 13. If from any

2pp' cos FPF' = p2 + p22 — 4c2.

point 0 two lines be drawn to the foci (or touching any confocal conic) meeting the conic in K, R'; S, S'; then (see also Ex. 15, Art. 231)

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It appears from the quadratic, by which the radius vector is determined (Art. 136), that the difference of the reciprocals of the roots will be the same for two values of 0, which give the same value to

(ac - g2) cos2 + 2 (ch — gf) cos 0 sin 0 + (be -ƒ2) sin2 0.

Now it is easy to see that A cos20 + 2H cos 0 sin 0 + B sin20 has equal values for any two values of 0, which correspond to the directions of lines equally inclined to the two represented by Ax2+ 2Hxy + By2 = 0. But the function we are considering becomes = 0 for the direction of the two tangents through 0 (Art. 147); and tangents to any confocal are equally inclined to these tangents (Art. 189). It follows from this example that chords which touch a confocal conic are proportional to the squares of the parallel diameters (see Ex. 15, Art. 231).

227. We give in this Article some examples on the parabola. The reader will have no difficulty in distinguishing those of the examples of the last Article, the proofs of which apply equally to the parabola.

Ex. 1. Find the coordinates of the intersection of the two tangents at the points x'y', x'y', to the parabola y2 = px. y' + y' yy' Ans. y= P

2 "

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Ex. 2. Find the locus of the intersection of the perpendicular from focus on tangent with the radius vector from vertex to the point of contact.

Ex. 3. The three perpendiculars of the triangle formed by three tangents intersect on the directrix (Steiner, Gergonne, Annales, XIX. 59; Walton, p. 119).

The equation of one of those perpendiculars is (Art. 32)

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The symmetry of the equation shows that the three perpendiculars intersect on the directrix at a height.

2y'y"y"" y′+y" +y""

y

+

D2

2

Ex. 4. The area of the triangle formed by three tangents is half that of the triangle formed by joining their points of contact (Gregory, Cambridge Journal, 11. 16 Walton, p. 137. See also Lessons on Higher Algebra, Ex. 12, p. 15).

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