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 the auxiliary circle), and the tangent at Fig. 20. that 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 the auxiliary circle, 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'. 22. We have investigated (Art. 10, Chap. IV.) the equation 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. We may hence obtain equations for determining the foci of the conic represented by the general equation of the second degree. For since the reciprocal of a conic with respect to a focus is a circle, it will follow from the above reasoning 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 − 2 W'fg) b2 + 2 (U'ƒ2 + Ugh — W'hf — V'fg) bc = (Vƒ2 + Ug2 — 2 W'fg) a2 + (Wg3 + Vh2 – 2 U'gh)c2 = = +2 (V'g'+Vhf U'fg- W'gh) ca (Wg2+Vh2 - 2 U'gh) b2 + (Uh2 + Wƒ2 − 2 V'hƒ) a2 +2 (W'h2+Wfg - V'gh — U'hf) ab, (Vb2+Wc2 + 2 U'bc) ƒ2—2 (V'c+W'b) ƒ (bg+ch)+U (bg+ch)2 = ( Wc2+Ua2 + 2 V'ca) g2 – 2 (W'a+U'c) g (ch+af)+V (ch + af)2 = = ( Ua2+Vb2 + 2 W'ab) h2— 2 ( U'b +V'a) h (af +bg) +W(af+bg)2, equations which, since af + bg+ch = 2▲, may also be written under the form = = (Ua2 + Vb2 + Wc2 + 2 U'bc +2 V'ca + 2 W'ab) ƒ2 +4A (V'c + W'b – Ua) ƒ+4U. A2 = (Ua2 + Vb2 + We2 + 2 U'bc + 2 V'ca + 2 W'ab) g2 + 4A (W'a + U'c - Vb) g + 4V.A2 (Ua2 + Vb2 + Wc2 + 2 U'bc + 2 V'ca + 2 W'ab)h2 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. VI., these equations reduce to (V'c+ Wb - Ua) ƒ + U▲ = (W'a + U'c - Vb) g + VA = which give the focus in that case. 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.; C,, C, respectively. Prove that a conic can be described touching the six straight lines ÂÂ ÂÂ ÂÂ BB, CC1, CC2 29 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. 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 line cut BC, CA, AB in D, E, F respectively, each of the angles AOD, BOE, COF is a right angle. Prove also that the polar of O with respect to this conic will always touch a conic, of which O is a focus. 10. OA, OB, are the common tangents to two conics having a common focus S, CA, CB are tangents at one of their points of intersection, BD, AE tangents intersecting CA, CB in D, E. Prove that S, D, E lie in the same straight line. 11. Any triangle is described, self-conjugate with regard to a given conic; prove that, if a conic be described, touching the sides of this triangle, and having the centre of the given conic as a focus, its axis-minor will be constant. 12. Prove that two ellipses which have a common focus, cannot intersect in more than two points. 23. Interesting results may sometimes be obtained by a double application of the method of reciprocal polars. Thus, Euc. III. 21 may be expressed under the form, "If a chord be drawn to a circle, subtending a constant angle at a fixed point O 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 in the curve, it always touches a conic having double contact with the given one. The forms which these theorems assume, when the constant angle is a right angle, are worthy of notice. |