Page images
PDF
EPUB

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.

tion

cut

Suppose then, that the curve, represented by the equa

ua2 + vß2 + wy2 + 2u'ẞy + 2v'ya + 2w'aß = 0,
BC in b, c,, CA in c2, α, AB in az,

[ocr errors]

37

[ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors]

Let h, h' be the respective distances of c2, a, from AB; g, g', those of a, b, 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 B0 in the equation of the conic section, bearing in mind that, when ẞ=0,

=

aa+cy = 2A.

This gives, for the determination of y, the equation

u (cy — 2A)2 + wa3y2 + 2av'y (2▲ — cy) = 0 ;

whence, by the theory of equations,

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

Hence, since Ac. Aα = Aα. Ab, we obtain

uc2+wa2 - 2v'ca va2 + ub2 - 2w'ab.

=

[merged small][merged small][ocr errors][subsumed][merged small]

The condition

gives

Cb1. Cc1 = Cc2. Ca2

wb2 + vc2 - 2u'bc = uc2 + wa2 - 2v'ca,

which also follows from the preceding two equations. Hence the equations

wb2+vc2-2u'bcuc2 + wa2 - 2v'ca va2 + ub2 - 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,

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

Substituting these values in the equation

ua2 + vß2 + wy2+2u' By + 2v'ya+2w'aß = 0,

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][subsumed][merged small][merged small]

(2u'bc — vc2 — wb2) aßy + (2v'ca — wa2 — uc2) bya

+(2w'ab — ub2 — va2) caß = 0,

which, if the conic be a circle, reduces to

aBy+bya + 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 P1, P2, P3 be the semi-diame

ters of the conic respectively parallel to the sides of the triangle of reference,

2

P12 (wb2+vc2-2u'bc) = p22 (uc2 + wa2 - 2v'ca)

[blocks in formation]

Hence, if two conics be similar and similarly situated, the values of the ratios denoted by

wb2+vc2 - 2u'bc: uc2 + wa2 - 2v'ca: va2 + ub2 — 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

Wg2+Vh2 - 2 U'gh + Uh2 +Wf2 − 2 V 'hf + Vƒ2 + Ug2 — 2 W'fg + 2 (U'ƒ2 + Ugh - V'fg - W'hf) cos A

+2 (V'g3 + Vhf — W'gh – U'fg) cos B

+2(W'h2+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.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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 line whatever, since, whatever l, m, n may be,

al+bm+cn− (bn+cm) cos A− (el+an) cos B−(am+bl) 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

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]
« PreviousContinue »