MAGNITUDES OF THE AXES OF THE CONIC.

93

'Ua2+ Vb2+Wc2+2 U'bc+2 V'ca+2 W'ab\

-(Va2

abc

(agh+bhf+cfg)=0.

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—(cl+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

21. To find the magnitude of the axes of the conic.

Let a, B, y be the co-ordinates of the centre; and, for shortness' sake, put

Then if r be the semi-diameter drawn from the centre to a, ß, y, we have (see Art. 3, Chap. 1)

(a cos A. x2+bcos B. y2+ccos, C. 23).............. (1).

Again, from the equation of the conic,

0 = $(α, ß, y) = $(ã+x, B+y, 7+ z)

=

B,

$ (a, B, y) + 2x (ua + w'ß + v'ỹ)

+2y (w'a+vß+u'ŋ) + 2z (v'a+u'ß + wỹ) +$ (x, y, z).

Now, by Art. 11 of the present chapter,

ua+w'ß + v'y _ w'a + vß+u'î _ v'a+u'ß+wy

a

_

Also, ax+by+cz = a (a− a) + b (B − B) + c (y − 7) = 0...(2) ;

:: 4 (x, y, z) = − ¢& (ā, B, Y),

or, ux2 + vy2+wz2 + 2u'yz +2v'zx + 2w'xy

Now the semi-axes are the greatest and least values of the semi-diameter. We have then to make

442

abc

r2 = a cos A.x2 + b cos B. y2+ c cos C.z3.

(4)

a maximum or minimum, x, y, z being connected by the relations (2) and (3).

Multiply (2) by the indeterminate multiplier 2μ, (4) by λ, adding them to (3), differentiating, and equating to zero the coefficients of each differential, we get

Multiplying these equations in order by x, y, z, and add

ing, we get

Substituting this value of λ in equations (5), and eliminating x, y, z from the equations combined with (2), we obtain the following quadratic for the determination of

1

22. To find the area of the conic.

1

In the above equation, the coefficient of is

abcs (a cos B cos C+b cos Ccos A + c cos A cos B),

Hence the product of the two values of 2 is

From the above investigation may be obtained the criterion which determines whether the conic be an ellipse or hyperbola. For, in the hyperbola, the two values of have opposite signs, hence the curve will be an ellipse or hyperbola according as

u, w', v', a

w', v, u', b

v', u', w. C

a, b, C, 0

is negative or positive; or according as

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

is positive or negative.

EXAMPLES.

1. Each angular point of a triangle is joined with each of two given points; prove that the six points of intersection of the joining lines with the opposite sides of the triangle lie in a conic.

2. A conic is described, touching three given straight lines and passing through a given point; prove that the locus of its centre is a conic.

Express, in geometrical language, the position of the given point relatively to the straight lines, in order that the locus of the centre may be a circle.

Also find the locus of the given point, in order that the locus of the centre may be a rectangular hyperbola.

3. Four circles are described, so that each of the four triangles, formed by each three of four given straight lines, is selfconjugate with respect to one of them; prove that the four circles have a common radical axis.

4. If A, B, C, A', B, C' be six points, such that the straight lines B'C', C'A', A'B' are the several polars of the points A, B, C, with respect to a given conic, prove that