CHAPTER IV.

ON THE CONIC REPRESENTED BY THE GENERAL EQUATION OF THE SECOND DEGREE.

1. WE may now proceed to the discussion of the general equation of the second degree, which we shall express under the form,

ua2+vß2+wy2+2u'By+2v'ya + 2w'aß = 0.

This we may write, for shortness, & (a, B, y) = 0.

This equation, as we have shewn (Art. 1, Chap. II.), represents a conic section.

2. To find the point in which a straight line, drawn in a given direction through a given point of the conic, meets the conic again.

Let f, g, h be the co-ordinates of the given point, a, B, y those of any other point whatever. Then, for all points of the straight line joining these two, the quantities

a-f, B-g, y-h,

will bear constant ratios to one another. Let these ratios be denoted by p: q: r, so that we have

To find where the line again meets the conic, we must substitute in the equation of the conic

f+ps for a, g+qs for ß, h+rs for y.

We thus get, arranging the result according to ascending powers of s,

Þ (f, g, h) + 2 {(up + w'q + v'r) ƒ +(w'p+vq+u'r) g

+(v'p+u'q+wr) h} s + 4 (p, q, r) s3 = 0.

The two roots of this equation, considered as a quadratic in s, determine the two points where the line meets the conic.

Now, since (f, g, h) is, by supposition, a point on the conic, it follows that (f, g, h) must be itself = 0. Hence, one of the two values of s, given by the above equation, will = 0, as ought to be the case, this value corresponding to the point f, g, h itself. The value of s, corresponding to the other point of intersection, will then be

- 2

(up+w'q+v'r) ƒ+ (w'p + vq+u'r) g+ (v'p+u'q+wr) h (p, q, r)

Hence, the values of a, ẞ, y, may be determined.

To this value of s, we shall hereafter have occasion to refer.

3. To find the equation of the tangent at a given point.

If the two points in which a straight line meets the conic be indefinitely close together, the value of s, investigated in Art. 2, must be = 0. This gives

(up+w'q + v'r) ƒ + (w'p+vq+u'r) g + (v'p + u'q + wr) h = 0,

or,

(uf+w'g+v'h) p+ (w'f+vg+u'h) q + (v'f+u'g+wh) r = 0. Hence, since, for every point on the line required,

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75

we get

(uf+w'g + v'h) a + (w'f+vg+u'h) B + (v'f+u'g+wh) y

=

uf2 + vg3 + wh3 + 2u'gh + 2v'hƒ + 2w'fg

= 0, since (f, g, h) is a point on the conic.

The tangent, therefore, at (f, g, h) is represented by the equation

(uf+w'g + v'h) a + (w'ƒ + vg+u'h) B + (v'f+u'g+wh) y = 0.

OBS. Those who are acquainted with the Differential Calculus will remark that this equation may be written thus,

4. To find the condition that a given straight line may

touch the conic.

Let the equation of the given straight line be

la + mẞ+ny = 0.

Let (f, g, h) be the co-ordinates of its point of contact; then, comparing this with the equation of the tangent just investigated, we see that we must have

uf+w'g+v'h _ w'f+vg+u'h _ v'f+u'g+wh

_ _

Representing each of these equivalent quantities by – k,

we shall have

Also, since (f, g, h) is a point on the given line,

Eliminating f, g, h, k, between (1), (2), (3), (4), we

obtain

as the necessary condition that the line (l, m, n) should touch the conic (f, g, h) = 0. Expanding the determinant, this may be written

(vw — u'3) 12 + (wu — v'2) m2 + (uv — w′2) n2 + 2 (v'w' — uu') mn

+ 2 (w'u' — vv') nl + 2 (u'v' — ww') Im = 0.

5. The coefficients of 12, m3, n2, 2mn, 2nl, 2lm, in the above equation, will be observed to be the several minors of the determinant

u, w',

w', v, u

v', u', w

They will frequently present themselves in subsequent investigations, and it will be convenient, therefore, to denote each by a single letter. We shall adopt the following notation:

=

v'w' - uu'=U', w'u' - vv'V', u'v'- ww' W'.

The condition of tangency investigated in Art. 4 may then be written,

Ul2 + Vm2 + Wn2 + 2 U'mn + 2 V'nl + 2 W'lm = 0,

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the same condition, it will be observed, as that which must hold, in order that the point (l, m, n) may lie on the conic

Ua2 + VB2 + Wy2+2 U'By + 2 V'ya +2 W'aß = 0.

6. To find the condition that the conic may be a parabola.

Since every parabola touches the line at infinity, the condition required will be obtained by writing a, b, c respectively in place of l, m, n, in the condition of tangency. This gives, as the necessary and sufficient relation among the coefficients,

or Ua+Vb2+ Wc+2 U'bc + 2 V'ca + 2 W'ab = 0.

7. To find the condition that the conic may break up into two straight lines, real or imaginary.

For this purpose it is necessary and sufficient that the expression (a, B, y) should break up into two factors. The condition for this has been shewn in Art. 9, Chap. III. to be

8. To find the equation of the polar of a given point.

If through a given point any straight line be drawn cutting a conic in two points, and at each point of section a tangent be drawn to the curve, the locus of the intersection of these tangents is the polar of the given point. We proceed to find the equation of the polar of (f, g, h).

Let f, g, h1; fa, 92, h, be the co-ordinates of the points in which any straight line drawn through (f, g, h) meets the