« PreviousContinue »
ON THE INTERSECTION OF CONICS, ON PROJECTIONS, AND ON
THE DETERMINATION OF A CONIC FROM FIVE GIVEN
1. We shall here say a few words on the subject of the intersection of two conics, as an acquaintance with this branch of the subject will be useful in future investigations.
Since every conic is represented by an equation of the second degree, it follows that any two conics intersect in four points, which may be (1) all real, (2) two real and two imaginary, or (3) all imaginary.
2. Through these four points of intersection three pairs of straight lines can be drawn. If the four points be called P, Q, R, S, the pairs of straight lines will be PQ and RS, PR and QS, PS and QR. If PQ and RS intersect in L, PR and QS in M, PS and QR in N, the points L, M, N are called (see Art. 15, Chap. II.) the vertices of the quadrangle PQRS. Also the three points L, M, N will form, with respect to every conic passing through the points P, Q, R, S, a conjugate triad; and therefore, each of them will have the same polar with respect to all such conics.
3. The equations of the pairs of lines PQ, RS, &c. (the sides and diagonals of the quadrangle) may be found as follows. Let the equations of the conics be $ (a, b, y) = ua* + vße + wory' + 2u'By+ 2v'rya + 20'aß= 0... (1), f(a, b, y)=pa? +qß* + rogo + 2p'Bry+20'ya +2r'aß = 0... (2); then every conic passing through their four points of intersection will be represented by an equation of the form
$(a, b, y) + ky (a, b, y) = 0 .....................(3).
If the left-hand member of this equation break up into two factors, the conic degenerates into two straight lines, real or imaginary. The condition that this should happen is
u+pk, w' to'k,
0 +qk u +p'k
= 0 ...........
w + rk
a cubic for the determination of k, of which the roots are either all real, or one real and two imaginary. If the roots be all real, the vertices of the quadrangle, which will be the centres of the several conics included in the form (3), will be all real. If one root only be real, then one vertex only of the quadrangle will be real. We proceed to consider how the reality of the vertices L, M, N depends upon that of the points P, Q, R, S.
4. First, suppose all the four points P, Q, R, S to be real, then it is clear that all the vertices will be real.
5. Next, let two of the points, P, Q, for example, be real, and R, S, imaginary.
Then, the line PR can have no other real point but P. For, if it had, it would itself become a real line, and we should have a real line cutting a real conic in one real and one imaginary point, which is impossible.
Hence the point M, which lies on PR, is imaginary. Similarly the line PS, and the point N, which lies on it, are imaginary. The real vertex must therefore be L, which lies on PQ.
We may observe that the line RS will be real. For the two lines PQ, RS, considered as one locus, will be represented by equation (3) when for k is substituted the real value corresponding to the point L. Hence the form of the expression $ (a, B, Y) +ky (a, b, y) answering to PQ, contains one real linear factor, and the other linear factor, which answers to RS, will therefore also be real.
6. Thirdly, let all the four points of intersection be imaginary. Then the three vertices will all be real.
For, by what has been shewn above, one vertex is necessarily so. Take this as the angular point A of the triangle of reference, and let its polar with respect to the two conics be taken for the side BC.
The point B being chosen arbitrarily, let its polar with respect to one of the conics be taken as AC. Then this conic may be represented by the equation
a +vB9+ wy= 0...............(1). Let the other be represented by
a® +9B2 + my +2p' By=0......... (2). Since the four points of intersection are imaginary, the roots of the quadratic
(2-v) B2 + (n − w) y? + 2p' Bry=0) will be imaginary. Hence (9-v) (r — w) >p.
(3). Now let (0, 9, h) be the co-ordinates of either vertex. Then, since it has the same polar with respect to both conics, the equations
vgB+ why=0, (29+p'k) B+(p'g+rh) y=0 will represent the same straight line, hence
99 + p'h_pʻg+rh
mine the vertices.
The two values of f, given by this equation, will deter
Now the roots of this equation are real or imaginary, as
(qw – rov)* + 4vwp'? 0.
That is, the vertices will necessarily be real, if v and w have the same signs. Suppose them, however, to have contrary signs, then, by (3),
qr-p? > qw+rv - vw; therefore multiplying both sides by – 4vw, which is a positive quantity,
4vw (qr-p") > 4vw (vw – qw – rv); .. (qw+ru)— 4vw (gr- p2) > (qw+ru)+4v?w* — 4vw (qw+ru)
> (qw + rv – 2vw)?
.. (qw - rv)? + 4vwp > 0. Hence, when the four points of intersection are imaginary, the vertices are in all cases real.
7. Suppose now that these vertices are taken as angular points of the triangle of reference. Let the conics be represented by the equations
ua+upa + wy* = 0, pas +232 + my = 0. B
y Then +
(qw – rull (ru – pro)a= + (po – quela are the equations of the several pairs of common chords of the two conics. Since two of the expressions
qw - rv, ru - pw, po — qu, must necessarily have the same sign, it follows that one pair at least of common chords is always real. The other two pairs will, as may easily be seen, be real or imaginary, according as the four points of intersection are, or are not, all real.
The above investigations are of use in the theory of projections, to which we now proceed.
8. DEF. The surface generated by a straight line of indefinite length, which always passes through a given fixed point, and always meets a given curve, the curve and point not lying in the same plane, is called a cone.
The fixed point is called the vertex, and will be denoted in this chapter by the letter V.
If a cone be cut by any two planes, either of the curves of section is said to be a projection of the other.
Also the two points in which any generating line is cut by two planes are said to be the projections, the one of the other. It
may easily be seen that the projection of any curve on a given plane coincides with the shadow of the curve which would be cast upon the plane by a luminous point coinciding with the vertex of the cone.
The projection of a point of intersection of any two curves will be a point of intersection of their projections.
The projection of any straight line will be a straight line; and that of any curve of the nth degree will be a curve of the nth degree. For since any straight line and curve of the nth degree intersect in n points, their projections will also intersect in n points.
9. If AB be any given straight line, and a cone be cut by any plane parallel to VAB, the projection of the line AB will be infinitely distant. Hence it is always possible so to project a figure, that the projection of any given straight line shall be removed to an infinite distance. This is called projecting the straight line to infinity.
10. Any quadrilateral may be projected into a parallelogram.
For, if ABCD be any quadrilateral, and the sides AB, CD be produced to meet in E, AD, BC in F, and the line EF projected to infinity, then, since the projections of AB, CD intersect at an infinite distance, they will be parallel to