CHAPTER IX.

MISCELLANEOUS PROPOSITIONS.

ON THE DETERMINATION OF A CONIC FROM FIVE GIVEN GEOMETRICAL CONDITIONS.

1. IF any five independent conditions be given, to which a conic is to be subject, each of these, expressed in algebraical language, will give an equation for the determination of the five arbitrary constants which the equation of the conic involves. Hence, five conditions suffice for the determination of the conic. It may, however, happen that some of the equations for the determination of the constants rise to a degree higher than the first, in such a case, the constants will have more than one value, and more than one conic may therefore be described, satisfying the required conditions, although the number will still be finite.

The geometrical conditions of most frequent occurrence are those of passing through given points and touching given straight lines, with such others as may be reduced to these. We proceed to consider how many conics may be described in each individual case.

2. Let five points be given.

In this case we have merely to substitute in the equation of the conic the co-ordinates of the several points for a, B, y; we shall thus obtain five simple equations for the determination of the constants, and one conic only will satisfy the given conditions.

3. Let four points and one tangent be given.

Take three of the points as angular points of the triangle of reference. Let f, g, h be the co-ordinates of the fourth

given point, la+mẞ + ny = 0, the equation of the given tangent. Let the equation of the conic be

Then for the determination of the ratios :μ:v, we have the equations

x2l2 + μ3m2 + v3n2 — 2μvmn – 2vλnl - 2xμlm = 0.

These equations will give two values for the ratios, and prove therefore that two conics can be described satisfying the required conditions.

4. Let three points and two tangents be given.

Take the three points as angular points of the triangle of reference. Let the two given tangents be represented by the equations

la + mB + ny = 0,

l'a + m'B + n'y = 0.

we have, for the determination of λ: :v, the equations

x2l2 + μ3m2 + v3n2 — 2μvmn 2vλnl

2λμίν = 0,

X3l12 + μ3m22 + v3n'2 — 2μvm'n' — 2vλn'l' — 2λμl'm' = 0,
λι mm

which, being both quadratics, give four values for each of the ratios, shewing that four conics may be described satisfying the given conditions.

5. Let two points and three tangents be given.

Take the three tangents as lines of reference, and let f, g, h; f', g', h', be the co-ordinates of the two given points.

Then, if the equation of the conic be

λ3a2 + μ3ß2 + v2y2 — 2μvßy — 2vλya – 2λμaß = 0,

we shall get, writing f, g, h; f', g', h', successively for a, ß, y, two quadratics for the determination of the ratios λ:μ:v, giving therefore four conics.

6. Let one point and four tangents be given.

Taking three of the tangents as lines of reference, the condition of touching the fourth given line gives a simple equation for the determination of the coefficients, and that of passing through the given point a quadratic. Hence, two conics may be described, satisfying the given conditions.

7. Let five tangents be given.

Taking three of the tangents as lines of reference, the condition of touching each of the others gives a simple equation for the determination of the constants, shewing that one conic only can be described satisfying these conditions.

The results of Arts. 5, 6, 7, may of course be deduced by the method of reciprocal polars, from those of Arts. 4, 3, 2.

8. Several other forms under which the data may be given, are reducible to a certain number of lines and points. Thus to have given a tangent and its point of contact is equivalent to having two points given, the points being indefinitely close together. Or, it may be regarded as equivalent to having two tangents given, these tangents being indefinitely nearly coincident. To have given that a conic is a parabola is equivalent to having a tangent given, since every parabola touches the line at infinity. To have given that it is a circle is equivalent to having two points given, since all circles intersect the line at infinity in the same two points. And this explains the reason why four circles can be described touching the sides of a given triangle, but only one circumscribed about it. So, to have given that a conic is similar and similarly situated to a given one is equivalent to having two points given. To have given an asymptote is equivalent to having two points given, for an asymptote may

be regarded as a tangent, the point of contact of which is given (at an infinite distance). To have given the direction of an asymptote is equivalent to having one point given, for this virtually determines the point in which the conic meets the line at infinity.

9. If it be given that three given points form a conjugate triad, this is equivalent to three conditions, as the equation of the conic, when these are taken as angular points of the triangle of reference, is of the form

ua2 + vẞ2 + wy2 = 0.

Two more conditions will therefore completely determine the conic. If these conditions be that the conic shall pass through two given points, or touch two given straight lines, or pass through one given point and touch one given straight line, one conic only can be drawn to satisfy these condi

tions.

We may observe that, if the above conic pass through the point (f, g, h), it also passes through the three points (-f, g, h), (f, g, h), (f, g, h), and that, if it touch the line (1, m, n), it also touches the lines (- l, m, n), (l, — m, n), (l, m, — n).

ON THE LOCUS OF THE CENTRE OF A SYSTEM OF CONICS WHICH SATISFY FOUR CONDITIONS, EXPRESSED BY PASSING THROUGH POINTS AND TOUCHING STRAIGHT LINES.

10. The locus of the centre of a conic, which passes through m points, and touches n straight lines, m + n being equal to four, will be a conic, in every case except two. We will consider the several cases in order.

11. Let the system pass through four points.

This is best treated by Cartesian co-ordinates.

Of the conics which can be described passing through four points, two are parabolas. Take, as co-ordinate axes, that diameter of each of these parabolas, the tangent at the ex

tremity of which is parallel to the axis of the other. Then the two parabolas will be represented by the equations

x2+2fy + h = 0...(1), y2 +2g'ah'0.............

.(2).

The system of conics is represented by the equation
x2 +λy2 + 2λg'x + 2fy + h + Xh′ =0.......................(3),

A being an arbitrary multiplier.

The centre is given by the equations

x+λg=0, xy+f=0.

Eliminating λ, we get for the locus of the centres

a conic, whose asymptotes are parallel to the axes of the parabolas (1) and (2).

If the four points form a convex quadrangle, the parabolas will be real, and the locus (4) an hyperbola. If the quadrangle be concave, the parabolas will be imaginary, and the locus of the centres an ellipse.

The curve (4) bisects the distance between each pair of the four points, and passes through the vertices of the quadrangle. This may be seen from geometrical considerations, for the three pairs of straight lines which belong to this system of conics, the vertices are respectively the centres.

From the form of the equation (3) we see that every conic of the system has a pair of conjugate diameters parallel to the axes of the parabola (1), (2); in other words to the asymptotes of (4).

λ = 1.

The conic of minimum eccentricity is obtained by making In this case, these are the equal conjugate diameters. If the axes of the parabolas be at right angles to one another, the four points lie on the circumference of a circle. The axes of every conic in (3) are then parallel to the coordinate axes, and (4) is a rectangular hyperbola.

If each of the four points be the orthocentre of the other