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Articles, that all circles pass through the same two imaginary points at infinity, and on that account can never intersect in more than two finite points, and that concentric circles touch each other in two imaginary points at infinity; and on that account can never intersect in any finite point. It will appear hereafter that a multitude of theorems concerning circles are but particular cases of theorems concerning conics which pass through two fixed points.

258. It is important to notice the form la+m3ß" = n2y2, which denotes a conic with respect to which a, B, y are the sides of a self-conjugate triangle (Art. 99). For the equation may be written in any of the forms

n3y2 — m3ß2 = l'a2 ; n3y* — l*a2 = m2ß2 ; l2x2 +m3ß2 = n'y3. The first form shows that ny + mß, ny - mẞ (which intersect in By) are tangents, and a their chord of contact. Consequently the point By is the pole of a. Similarly from the second form ya is the pole of B. It follows, then, that aß is the pole of y; and this also appears from the third form, which shows that the two imaginary lines la + mẞ √(−1) are tangents whose chord of contact is y. Now these imaginary lines intersect in the real point aß, which is therefore the pole of y; although being within the conic, the tangents through it are imaginary.

It appears, in like manner, that

aa +2haß + b3 = c

denotes a conic, such that aß is the pole of y; for the left-hand side can be resolved into the product of factors representing lines which intersect in aß.

COR. If a2+ m2ß2 = n2y2 denote a circle, its centre must be the intersection of perpendiculars of the triangle aẞy. For the perpendicular let fall from any point on its polar must pass through the centre.

258* (a). If x = 0, y=0 be any lines at right angles to each other through a focus, and y the corresponding directrix, the equation of the curve is

a particular form of the

x2 + y2 = e2y2,

equation of Art. 258. Its form shows that the focus (xy) is the pole of the directrix y, and that the

*This Article was numbered 279 in the previous editions.

polar of any point on the directrix is perpendicular to the line joining it to the focus (Art. 192); for y, the polar of (xy) is perpendicular to x, but x may be any line drawn through the focus.

The form of the equation shows that the two imaginary lines + y2 are tangents drawn through the focus. Now, since these lines are the same whatever y be, it appears that all conics which have the same focus have two imaginary common tangents passing through this focus. All conics, therefore, which have both foci common, have four imaginary common tangents, and may be considered as conics inscribed in the same quadrilateral. The imaginary tangents through the focus (x2 + y2 = 0) are the same as the lines drawn to the two imaginary points at infinity on any circle (see Art. 257). Hence, we obtain the following general conception of foci: "Through each of the two imaginary points at infinity on any circle draw two tangents to the conic; these tangents will form a quadrilateral, two of whose vertices will be real and the foci of the curve, the other two may be considered as imaginary foci of the curve."

Ex. To find the foci of the conic given by the general equation. We have only to express the condition that x - x' + (y - y') √(− 1) should touch the curve. Substituting then in the formula of Art. 151, for A,, v respectively, 1, √(− 1), − {x′ + y′ √(− 1)}; and equating separately the real and imaginary parts to cypher, we find that the foci are determined as the intersection of the two loci

C (x2 - y2) +2Fy - 2Gx + A- B=0, Cxy - Fx - Gy + H = 0, which denote two equilateral hyperbolas concentric with the given conic. Writing the equations

(CxG)

-

(Cy - F)2 G2 AC - (F2 – BC) = ▲ (a - b), (CxG) (Cy- F) = FG - CH = Ah;

the coordinates of the foci are immediately given by the equations

(Cx − G)2 = }▲ (R + a − b) ; (Cy − F)2 = }▲ (R + b − a),

-

where ▲ has the same meaning as at p. 153, and R as at p. 158. If the curve is a parabola, C=0, and we have to solve two linear equations which give

(F2 + G2) x = FH + } (A − B) G ; (F2 + G2) y = GH + } (B − A) F. 259. We proceed to notice some inferences which follow on interpreting, by the help of Art. 34, the equations we have already used. Thus (see Arts. 122, 123) the equation ay = kß implies that the product of the perpendiculars from any point of a conic on two fixed tangents is in a constant ratio to the square of the perpendicular on their chord of contact.

The equation ay = kß8, similarly interpreted, leads to the

important theorem: The product of the perpendiculars let fall from any point of a conic on two opposite sides of an inscribed quadrilateral is in a constant ratio to the product of the perpendiculars let fall on the other two sides.

From this property we at once infer, that the anharmonic ratio of a pencil, whose sides pass through four fixed points of a conic, and whose vertex is any variable point of it, is constant. For the perpendicular

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but the right-hand member of this equation is constant, while the left-hand member is the anharmonic ratio of the pencil OA, OB, OC, OD.

The consequences of this theorem are so numerous and important that we shall devote a section of another chapter to develope them more fully.

260. If S=0 be the equation to a circle, then (Art. 90) S is the square of the tangent from any point xy to the circle; hence S-kaẞ=0 (the equation of a conic whose chords of intersection with the circle are a and B) expresses that the locus of a point, such that the square of the tangent from it to a fixed circle is in a constant ratio to the product of its distances from two fixed lines, is a conic passing through the four points in which the fixed lines intersect the circle.

This theorem is equally true whatever be the magnitude of the circle, and whether the right lines meet the circle in real or imaginary points; thus, for example, if the circle be infinitely small, the locus of a point, the square of whose distance from a fixed point is in a constant ratio to the product of its distances from

two fixed lines, is a conic section; and the fixed lines may be considered as chords of imaginary intersection of the conic with an infinitely small circle whose centre is the fixed point.

261. Similar inferences can be drawn from the equation S-ka = 0, where S is a circle. We learn that the locus of a point, such that the tangent from it to a fixed circle is in a constant ratio to its distance from a fixed line, is a conic touching the circle at the two points where the fixed line meets it; or, conversely, that if a circle have double contact with a conic, the tangent drawn to the circle from any point on the conic is in a constant ratio to the perpendicular from the point on the chord of contact.

In the particular case where the circle is infinitely small, we obtain the fundamental property of the focus and directrix, and we infer that the focus of any conic may be considered as an infinitely small circle, touching the conic in two imaginary points situated on the directrix.

262. In general, if in the equation of any conic the coordinates of any point be substituted, the result will be proportional to the rectangle under the segments of a chord drawn through the point parallel to a given line.*

For (Art. 148) this rectangle

=

a cos'+2h cose sine+b sin'0

9

where, by Art. 134, c' is the result of substituting in the equation the coordinates of the point; if, therefore, the angle be constant, this rectangle will be proportional to c'.

Ex. 1. If two conics have double contact, the square of the perpendicular from any point of one upon the chord of contact is in a constant ratio to the rectangle under the segments of that perpendicular made by the other.

Ex. 2. If a line parallel to a given one meets two conics in the points P, Q, p, q, and we take on it a point 0, such that the rectangle OP. OQ may be to Op. Oq in a constant ratio, the locus of O is a conic through the points of intersection of the given conics.

Ex. 3. The diameter of the circle circumscribing the triangle formed by two bb" tangents to a central conic and their chord of contact is ; where b', b' are the Р semi-diameters parallel to the tangents, and p is the perpendicular from the centre on the chord of contact. [Mr. Burnside].

This is equally true for curves of any degree.

It will be convenient to suppose the equation divided by such a constant that the result of substituting the coordinates of the centre shall be unity. Let t', t" be the lengths of the tangents, and let S' be the result of substituting the coordinates of their intersection; then

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But also if be the perpendicular on the chord of contact from the vertex of the triangle, it is easy to see, attending to the remark, Note, p. 154,

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But the left-hand side of this equation, by Elementary Geometry, represents the diameter of the circle circumscribing the triangle.

Ex. 4. The expression (Art. 242) for the radius of curvature may be deduced if in the last example we suppose the two tangents to coincide, in which case the diameter of the circle becomes the radius of curvature (see Art. 398); or also from the following theorem due to Mr. Roberts: If n, n' be the lengths of two intersecting normals; p, p' the corresponding central perpendiculars on tangents; b' the semi-diameter parallel to the chord joining the two points on the curve, then np + n'p' = 2b′2. For if S' be the result of substituting in the equation the coordinates of the middle point of the chord,, ' the perpendiculars from that point on the tangents, and 28 the length of the chord, then it can be proved, as in the last example, that B2 = b2S', = pS', w' = p'S', and it is very easy to see that nw + n'w' = 282.

263. If two conics have each double contact with a third, their chords of contact with the third conic, and a pair of their chords of intersection with each other, will all pass through the same point, and will form a harmonic pencil.

Let the equation of the third conic be S=0, and those of the first two conics,

S+L=0, S + M2 = 0.

Now, on subtracting these equations, we find L - M2 = 0, which represents a pair of chords of intersection (L+M=0) passing through the intersection of the chords of contact (L and M), and forming a harmonic pencil with them (Art. 57).

Ex. 1. The chords of contact of two conics with their common tangents pass through the intersection of a pair of their common chords. This is a particular case of the preceding, S being supposed to reduce to two right lines.

Ex. 2. The diagonals of any inscribed, and of the corresponding circumscribed quadrilateral, pass through the same point, and form a harmonic pencil. This is also a particular case of the preceding, S being any conic, and S + L2, S + M2 being supposed to reduce to right lines. The proof may also be stated thus: Let t1, t2, C1; t3. t, ca be two pairs of tangents and the corresponding chords of contact. In other words, c1, c2 are diagonals of the corresponding inscribed quadrilateral. equation of S may be written in either of the forms

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