FOCAL PROPERTIES.

123

Take, for example, Euc. III. 21. This may be expressed as follows: "If three points be taken on the circumference of a circle, two fixed and the third moveable, the straight lines joining the moveable point with the two fixed points, make a constant angle with one another." This will be reciprocated into "If three tangents be drawn to a conic section, two fixed and the third moveable, the portion of the moveable tangent intercepted between the two fixed ones, subtends a constant angle at the focus." This angle will be found, by reciprocating Euc. III. 20, to be the complement of onehalf of the angle subtended at the focus by the portion of the corresponding directrix intercepted between the two fixed tangents.

Again, it is easy to see that "if a circle be described touching two concentric circles, its radius will be equal to half the sum, or half the difference, of the radii of the given circles, and the locus of its centre will be a circle, concentric with the other two, and of which the radius is half the difference, or half the sum, of the radii of the two given circles."

Hence we deduce the following theorem. "If two conics have a common focus and directrix, and their latera-recta be 27, 27', and another conic, having the same focus, be described 4ll' so as to touch both of them, its latus-rectum will be

はじ and the envelope of its directrix will be a conic, having the same focus and directrix as the given conics, and of which 4ll' 12 1 = 1'°

the latus-rectum is

Again, take the ordinary definition of an ellipse, that it is the locus of a point, the sum of the distances of which from two fixed points is constant. This is equivalent to "the sum of the distances from either focus, of the points of contact of two parallel tangents, is constant."

The reciprocal theorem will be, "If a system of chords be drawn to a circle, passing through a given point, and, at the extremities of any chord, a pair of tangents be drawn to the circle, the sum of the reciprocals of the distances of these tangents from the fixed point is constant."

The known property of a circle, that "two tangents make equal angles with their chord of contact" will be found, when transformed by the method now explained, to be equivalent to the theorem that "if two tangents be drawn to a conic from an external point, the portions of these tangents, intercepted between that point and their points of contact, subtend equal angles at the focus." From the fact that "all circles intersect in two imaginary points at infinity," we learn that "all conics, having a common focus, have a common pair of imaginary tangents passing through that focus." And, more generally, we may say that all similar and similarly situated conics reciprocate into a system of conics having two common tangents.

31. Two points, on a curve and its reciprocal, are said to correspond to one another when the tangent at either point is the polar of the other point. Two tangents are said to correspond when the point of contact of either is the pole of the other.

The angle between the radius vector of any point (drawn from the centre of reciprocation), and the tangent at that

Y

Fig. 20.

point, is equal to the angle between the radius vector of, and tangent at, the corresponding point of the reciprocal

curve.

For, if P be the given point, PY the tangent at P, and S the centre of reciprocation, and SY be perpendicular to PY; and if P' be the pole of PY, and P'Y' the polar of P, then

P' will lie on SY, produced if necessary; and if SY' be perpendicular to P'Y', SY' will pass through P. Hence, since SP, PY, are respectively perpendicular to P'Y', SP', it follows that the angle SPY is equal to the angle SP'Y'.

equa

32. We have investigated (Art. 10, Chap. IV.) the tion of the two tangents drawn to a conic from any given point (f, g, h). If in the right-hand member of that equation we substitute for 0, w (aa+bB+cy)2, w being an arbitrary constant, we shall obtain the general equation of all conics of which these lines are asymptotes. Now, since the asymptotes of the reciprocal conic with respect to (f, g, h), are respectively at right angles to the two tangents drawn from (f, g, h), it follows that the family of conics thus obtained will be similar in form to the reciprocal conic.

33. To find the co-ordinates of the foci of the conic represented by the general equation of the second degree.

Since the reciprocal of a conic with respect to a focus is a circle, it will follow from Art. 32 that the family of conics obtained as above must, if (f, g, h) be a focus, be circles also. Applying the conditions for a circle investigated in Art. 14, Chap. IV., it will be found that the terms involving a disappear of themselves, and our conditions assume the form

or

(Uh2+Wƒ2 — 2V'hƒ) c2 + (Vƒ2+Ug2 — 2 W'ƒg) b2

+ 2 (U'ƒ2+Ugh — W'hf — V'fg) bc

= ( Vƒ2 + Ug2 − 2 W'fg) a2 + (Wg2 + Vh2 − 2 U'gh) c2

+ 2 (V'g2+Vhf -U'fg-W'gh) ca

=(Wg2+Vh2-2U'gh) b2 + (Uh2 + Wƒ2 — 2 V'hf) a2

+2(W'h2+Wfg-V'gh-U'hf) ab,

(Vb2+Wc2+2U'bc)ƒ3—2 (V'c+W'b) ƒ (bg+ch)+U(bg+ch)3 =(Wc2+Ua2+2V'ca) g'-2 (W'a+U'c) g (ch+af)+V(ch+af)2 = (Ua2+ Vb2+2W'ab) h'-2(U'b+V'a)h (af+bg)+W(af+bg)3,

equations which, since af+bg+ch = 2A, may also be written under the form

(Ua2 + Vb2+Wc3 +2 U'bc + 2 V'ca + 2 W'ab) ƒ2

— 4A (V'c+W'b+Ua) ƒ+4U.▲2

= (Ua2 + Vb2 + We2+2 U'bc + 2 V'ca + 2 W'ab) g'

=

(Ua2+Vb2+Wc+2U'bc + 2 V'ca + 2 W'ab) h2

−4▲ (U'b + V'a + Wc) h +4 W. A2.

The equations, together with

af + bg+ch = 2A,

⚫ determine the co-ordinates of the foci. It will be seen that they give four values of f, g, h, two of which are real, two imaginary.

If the conic be a parabola, then, applying the condition of Art. 6, Chap. IV., these equations reduce to

(V'c+W'b+Ua) ƒ— U▲ = (W'a+U'c+Vb) g− V▲

=

= (U'b + V'a + Wc).h — WA,

which give the focus in that case.

If the equation

ux2 + vy3 + wz2 + Qu'yz + 2v′zx + 2w'xy = 0,

be expressed in triangular co-ordinates, we get, for the coordinates of the foci, the equations

(U+V+W+2U'+2V'+2W') ƒ2−2 (V'+W'+U)ƒ+U

a2

(U+V+W+2U'+ 2 V'+2 W') g2 − 2 (W'+U'+V)g+V

(U+V+W+2U '+ 2 V'+2 W') h2 − 2 ( U '+ V'+W)h+W

2 c2

or, if the conic be a parabola,

2 (V'+ W'+U)ƒ— U _ 2 (W'+ U'+ V) g − V

a2

b2

2 (U'+ V'+ W) h – W

c2

34. Interesting results may sometimes be obtained by a double application of the method of reciprocal polars. Thus, the theorem that "the angle in a semicircle is a right angle' may be expressed in the form that "every chord of a circle, which subtends a right angle at a given point of the curve, passes through the centre." Reciprocating this with respect to the given point, we get

"The locus of the point of intersection of two tangents to a parabola at right angles to one another, is the directrix." Now, reciprocate this with respect to any point whatever, and we find that

"Every chord of a conic which subtends a right angle at a given point on the curve, passes through a fixed point."

Again, take Euc. III. 21. This may be expressed under the form "If a chord be drawn to a circle subtending a constant angle at a fixed point 0 on its circumference, it always touches a concentric circle." Reciprocating this theorem with respect to Ọ, we get "If two tangents be drawn to a parabola containing a constant angle, the locus of their point of intersection will be a conic, having a focus and directrix in common with the given parabola. Reciprocate this, with respect to any point whatever, and we get, "If a chord be drawn to a conic, subtending a constant angle at a given point on the curve, it always touches a conic having double constant with the given one.'

EXAMPLES.

1. Having given a focus and two points of a conic section, prove that the locus of the point of intersection of the tangents at these points will be two straight lines, passing through the focus, and at right angles to each other.

2. Prove that four conics can be described with a given focus and passing through three given points, and that the latus-rectum