« PreviousContinue »
In other words, if a rectangular hyperbola be so described that each angular point of a given triangle is the pole, with respect to it, of the opposite side, it will pass through the centres of the four circles which touch the three sides of the triangle.
14. To investigate the conditions that the general equation of the second degree shall represent a circle.
The property of the circle, which we shall assume as the basis of our investigation, is the following: that if, through any point, chords be drawn cutting a circle, the rectangle, contained by their segments, is invariable.
Suppose then, that the curve, represented by the equation
ua+ vß? + way + 2u'By + 2v'rya + 2w'aß =0, cut BC in by, C7, CA in c,, Qg, AB in ag, bz,
then, if this curve be a circle, Ac,. Aa, = Aag. Abg, Bag. Bbg = Bb,. Bcn, Cb,. Cc, = Cc . Ca,,
Let h, h' be the respective distances of c,, a, from AB; 9, 9', those of ag, bę from AC: then, multiplying the first of the above three equations by sin A, we get
hh' = gg' Now h, h' are the two values of y obtained by putting B=0 in the equation of the conic section, bearing in mind that, when ß=0,
aa + cy = 24.
u (cy — 24) + wa’ya + 2av'ry (2A – cy) = 0; whence, by the theory of equations,
uca + wa’ — 2v'ca'
Hence, since Acg. Aa, = Aag. Abg, we obtain
uc + wa’ – 2v'ca=va + ub? – 2w'ab. Similarly, from the condition
Bag. Bb, = Bb. Bc
The condition Cb,. Cc, = Cc . Ca gives wb + vc - 2u'bc = uc + wa - 2v'ca, which also follows from the preceding two equations. Hence the equations
w62 + vc — 2u'bc=uc + wa — 2v'ca = va’ + ub? – 2w'ab are necessary conditions that the given equation should represent a circle; and, since they are two in number, they are sufficient.
15. To determine the intersection of a circle with the line at infinity. Since, at every point in the line at infinity,
ao + b3 + cy=0, we shall have
ογα + βαβ
Substituting these values in the equation
ua+ vß2 + ways + 2u' By + 2v'rya +2w'aß= 0,
aß = 0;
b or, multiplying by abc, (2u'bc — vc? – wb) aßy + (2v'ca – wa? – uc) brya
+ (2w'ab – ub— va“) caß = 0, which, if the conic be a circle, reduces to
aßry + brya + caß = 0, shewing that every circle intersects the line at infinity in the same two points as the circle described about the triangle of reference; that is, all circles intersect the line at infinity in the same two points. These points are, of course, imaginary.
16. It may be shewn, by a geometrical investigation similar to that in Art. 14, that if Pas Pes Po be the semi-diame
ters of the conic respectively parallel to the sides of the triangle of reference, pi? (wb? + vc? – 2u'bc) = p. (uc? + wa’ – 2v'ca)
= pz (va+ ub? – 2w'ab). Hence, if two conics be similar and similarly situated, the values of the ratios denoted by
wb2 + vc — 2u'bc : uca + wa— 2v'ca : va? + ub? — 2w'ab must be the same for both.
Hence, also, by reasoning similar to that employed in Art. 15, it follows that all conics, similar and similarly situated to each other, intersect in the same two points in the line at infinity.
These points will be real, coincident, or imaginary, according as the conics are hyperbolas, parabolas, or ellipses.
If the conics, in addition to being similar and similarly situated, are also concentric, they will touch one another at the two points where they meet the line at infinity.
17. We have investigated, in Art, 10, the equation of the pair of tangents drawn to the conic from a given point f, g, h). If these two tangents be at right angles to one another, they may be regarded as the limiting form of a rectangular hyperbola, and must therefore satisfy the equation investigated in Art. 13. This, therefore, gives as the locus of the intersection of two tangents at right angles to one another Wg” + Vh? – 2U'gh + Uh® +Wf2 – 2V'hf + Vf® + Ugo – 2 W fg
+2 (U'f? + Ugh – V'fg – W'hf) cos A
+2 (W'h* + Wfg – U'hf – V'gh) cos C=0. This may be shewn (see Art. 15) to represent a circle, as we know ought to be the case.
This equation may also be expressed in the following form
V + W + 2 U'cos A W+U+2V'cos B (af + bg + ch)
9 5 UIV+2W'cos C +
If the conic be a parabola, then (see Art. 6) this breaks up into two factors, one of which is the line at infinity; and the other must represent the directrix, since that is the locus of the point of intersection of two tangents to a parabola at right angles to one another.
The appearance of the line at infinity as a factor in the result in this case may be explained as follows: Every parabola touches the line at infinity, and this line also satisfies the algebraical condition of being perpendicular to any linė whatever, since, whatever l, m, n may be, al+bm+cn-(bn+cm) cos A-(el+an) cos B-(am +6l) cos C=0, identically.
It therefore will form a part of the locus of the intersection of two tangents at right angles to one another, the two tangents being the line at infinity itself, and any other tangent whatever.
The directrix of the parabola is therefore represented by the equation V + W + 2 U'cos A W +U+2 V'cos B at