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 PY, 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'.

32. We have investigated (Art. 10, Chap. IV.) the equation of the two tangents drawn to a conic from any given point (f, g, h). In the right-hand member of that equation we substitute for 0, w (aa +bB+cy), 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'hf) c2 + (Vƒ2 + Ug2 – 2W'ƒg) b2

+ 2 (U'ƒ2 + Ugh — W'hf — V'fg) bc

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

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

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

+2(Wh2+Wfg-V'gh-Uhfab,

(Vb2+Wc3+2U′bc) ƒ2−2(V'c+W'b)f(bg+ch) + U(bg+ch)2 =(Wc2+Ua2+2V'ca)g'-2 (W'a+U'c) g (ch+af)+V(ch+af)2

=

= (Ua2+ Vb2+2W'ab) h3—2 (U'b+ V'a)h (af+bg)+W(af+bg)2,

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

under the form

2

(Ua2 + Vb2 + We2+2U'bc+2V'ca + 2 W'ab) ƒ2

− 4A (V'c + W′b + Ua) ƒ + 4U. A2

= (Ua2 + Vb2 + Wc2 + 2Ubc + 2 V'ca + 2 W'ab) g2

=

− 4A (W'a + U'c + Vb) g + 4 V. A3

= (Ua2 + Vb2 + Wc2 + 2Ubc + 2 V'ca + 2 W'ab) h2

- 4▲ (U'b + Va+ 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+ Wb+Ua)ƒ— UA = (W'a+Uc+ Vb) g − VA = (U'b + V'a+ Wc) h – WA,

=

which give the focus in that case.

If the equation

ux2 + vy2+wz2 + 2u'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'+2 V'+2W') ƒ2−2 (V'+W'+U)ƒ+U

a2

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

b2

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

or, if the conic be a parabola,

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

a2

=

62

2 (U'V'+W) h - W

c*

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 O, 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 contact 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 of one of these is equal to the sum of the latera-recta of the other three.

3. On a fixed tangent to a conic are taken a fixed point A, and two moveable points P, Q, such that AP, AQ subtend equal angles at a fixed point 0. From P, Q are drawn two other tangents to the conic, prove that the locus of their point of intersection is a straight line.

4. Two variable tangents are drawn to a conic section so that the portion of a fixed tangent, intercepted between them, subtends a right angle at a fixed point. Prove that the locus of the point of intersection of the variable tangents is a straight line.

If the fixed point be a focus, the locus will be the corresponding directrix.

5.

Chords are drawn to a conic, subtending a right angle at a fixed point; prove that they all touch a conic, of which that point is a focus.

6. Three given straight lines BC, CA, AB are intersected by two other given straight lines in A,, A.; B,, B.; C1, C, respectively. Prove that a conic can be described touching the six straight lines AA, AA, BB, BB, CC CC.

7. A, B, C, S are four fixed points, SD is drawn perpendicular to SA, intersecting BC in D, SE perpendicular to SB, intersecting CA in E, SF perpendicular to SC, intersecting AB in F. Prove that D, E, F lie in the same straight line.

Prove also that the four conics which have S as a focus, and which touch the three sides of the several triangles ABC, AEF, BFD, CDE have their latera-recta equal.

8. Two conics are described with a common focus and their corresponding directrices fixed; prove that, if the sum of the reciprocals of their latera-recta be constant, their common tangents will touch a conic section.

9. A conic is described touching three given straight lines BC, CA, AB, so that the pair of tangents drawn to it from a given point 0, are at right angles to each other. Prove that it will always touch another fixed straight line; and that, if this straight