it must be remarked that by a tangent we mean in general a line which meets the curve in two coincident points; if then the curve reduce to two right lines, the only line which can meet the locus in two coincident points is the line drawn to the point of intersection of these right lines, and since two tangents can always be drawn to a curve of the second degree, both tangents must in this case coincide with the line to the point of intersection.

148. If through any point O two chords be drawn, meeting the curve in the points R', R", S', S", then the ratio of the rectangles OR'.OR" will be constant, whatever be the position of the point 0, OS'.OS" provided that the directions of the lines OR, OS be constant. For, from the equation given to determine p in Art. 136, it appears that

But this is a constant ratio: for a, h, b remain unaltered when the equation is transformed to parallel axes through any new origin (Art. 134), and 0, 0' are evidently constant while the direction of the radii vectores is constant.

The theorem of this Article may be otherwise stated thus: If through two fixed points O and O' any two parallel lines OR

and O'p be drawn, then the ratio of the rectangles

be constant, whatever be the direction of these lines. For, these rectangles are

с

OR'. OR"

will

O'p'. O'p"

a cos 0 + 2h cose sin 0 + b sin'0' a cos20+2h cos e sin ◊ +b sin20 ' (c' being the new absolute term when the equation is transferred to O' as origin); the ratio of these rectangles =

fore, independent of 0.

C ==

and is, there

This theorem is the generalization of Euclid III. 35, 36.

U

149. The theorem of the last Article includes under it several particular cases, which it is useful to notice separately.

I. Let O' be the centre of the curve, then O'p' = O'p" and the quantity O'p'. O'p" becomes the square of the semidiameter parallel to OR'. Hence, The rectangles under the segments of two chords which intersect are to each other as the squares of the diameters parallel to those chords.

II. Let the line OR be a tangent, then OR' = OR", and the quantity OR'.OR" becomes the square of the tangent; and, since two tangents can be drawn through the point 0, we may extract the square root of the ratio found in the last paragraph, and infer that Two tangents drawn through any point are to each other as the diameters to which they are parallel.

III. Let the line 00' be a diameter, and OR, O'p parallel to` its ordinates, then OR' OR" and O'p' O'p". Let the diameter

=

=

meet the curve in the points A, B, then

OR2

O'p*

=

A0.0B AO.O'B'

Hence, The squares of the ordinates of any diameter are proportional to the rectangles under the segments which they make on the diameter.

150. There is one case in which the theorem of Article 148 becomes no longer applicable, namely, when the line OS is parallel to one of the lines which meet the curve at infinity; the segment OS" is then infinite, and OS only meets the curve in one finite point. We propose, in the present Article, to inquire OS' whether, in this case, the ratio will be constant. OR'.OR"

Let us, for simplicity, take the line OS for our axis of x, and OR for the axis of y. Since the axis of x is parallel to one of the lines which meet the curve at infinity, the coefficient a will = 0 (Art. 138, Ex. 4), and the equation of the curve will be of the form 2hxy+by2+2gx + 2fy + c = 0.

OS'

Making y = 0, the intercept on the axis of x is found to be

==

с

2g

; and making x=0, the rectangle under the inter

Now, if we transform the axes to any parallel axes (Art. 134), b will remain unaltered, and the new g=hy' + g.

Now, if the curve be a parabola, h=0, and this ratio is constant; hence, If a line parallel to a given one meet any diameter (Art. 142) of a parabola, the rectangle under its segments is in a constant ratio to the intercept on the diameter.

If the curve be a hyperbola, the ratio will only be constant while y' is constant; hence, The intercepts made by two parallel chords of a hyperbola, on a given line meeting the curve at infinity, are proportional to the rectangles under the segments of the chords.

*151. To find the condition that the line λx+μy+v may touch the conic represented by the general equation. Solving for y from λx+μy+v=0, and substituting in the equation of the conic; the abscissæ of the intersections of the line and curve are determined by the equation

(aμ2 — 2hλμ + bx2) x2 + 2 (gμ2 – hμv − ƒμλ +bλv) x

+ ( c − 2 furt br*)=0. The line will touch when the quadratic has equal roots, or when (aμ2 — 2hλμ+bλ2) (cμ3 — 2ƒμv + bv2) = (gμ2 – hμv — fμλ + bλv)2. Multiplying out, the equation proves to be divisible by μ, and

becomes

(bc −ƒ2) λ2 + (ca - g3) μ2 + (ab − h2) v2 + 2 (gh − aƒ) μv

+ 2 (hf −bg) vλ + 2 (fg − ch) λμ = 0.

We shall afterwards give other methods of obtaining this equation, which may be called the tangential equation of the curve. We shall often use abbreviations for the coefficients, and write the equation in the form

Ax2+ Bμ2 + Cv2 + 2 Fμv + 2 Gvλ + 2Hλμ = 0.

The values of the coefficients will be more easily remembered by

the help of the following rule. Let A denote the discriminant of the equation; that is to say, the function

whose vanishing is the condition that the equation may represent right lines. Then A is the derived function formed from ▲, regarding a as the variable; and B, C, 2F, 2G, 2H are the derived functions taken respectively with regard to b, c, f, g, h. The coordinates of the centre (given Art. 140) may be written

G F
C' C

MISCELLANEOUS EXAMPLES.

Ex. 1. Form the equation of the conic making intercepts λ, λ', μ, μ' on the axes. Since if we make y = 0, or x = 0 in the equation, it must reduce to

the equation is

x2 (^\ + λ) x + λλ′ = 0, y2 — (μ + μ”) y + μμ' = 0 ;

e

μμα + 2hy + λλ' ? - μμ' (λ + λ') α - λλ' (μ + μ*)y + λλόμμ' = 0,

x

and h is undetermined, unless another condition be given. Thus two parabolas can be drawn through the four given points; for in this case

h = + (λλ'μμ").

Ex. 2. Given four points on a conic, the polar of any fixed point passes through a fixed point. We may choose the axes so that the given points may lie two on each axis, and the equation of the curve is that found in Ex. 1. But the equation of the polar of any point x'y' (Art. 145) involves the indeterminate h in the first degree, and therefore passes through a fixed point.

Ex. 3. Find the locus of the centre of a conic passing through four fixed points. The centre of the conic in Ex. 1 is given by the equations

2μμ'x + 2hy — μμ' (λ + λ') = 0, 2XX'y + 2hx − XX' (μ + μ') = 0 ;

whence eliminating the indeterminate h, the locus is

2μμα - 2λλ'ψ - μμ' (λ + λ') π + λλ' (μ + μ" y = 0,

a conic passing through the intersections of each of the three pairs of lines which can be drawn through the four points, and through the middle points of these lines. The locus will be a hyperbola when X, X' and μ, μ' have either both like, or both unlike signs; and an ellipse in the contrary case. Thus it will be an ellipse when the two points on one axis lie on the same side of the origin, and on the other axis, on opposite sides. In other words, when the quadrilateral formed by the four given points has a re-entrant angle. This is also geometrically evident: for a quadrilateral with a re-entrant angle evidently cannot be inscribed in a figure of the shape of the ellipse or parabola. The circumscribing conic must therefore always be a hyperbola, so that some vertices may lie in opposite branches. And since the centre of a hyperbola is never at infinity, the locus of centres is in this case an ellipse. In the other case, two positions of the centre will be at infinity, corresponding to the two parabolas which can be described through the given points.

CHAPTER XI.

EQUATIONS OF THE SECOND DEGREE REFERRED TO THE
CENTRE AS ORIGIN.

152. In investigating the properties of the ellipse and hyperbola, we shall find our equations much simplified by choosing the centre for the origin of co-ordinates. If we transform the general equation of the second degree to the centre as origin, we saw (Art. 140) that the coefficients of x and y will 0 in the transformed equation, which will be of the form

ax2+2hxy+by2 + c' = 0.

It is sometimes useful to know the value of c' in terms of the coefficients of the first given equation. We saw (Art. 134) that c' = ax2 + 2hx'y' + by'2 + 2gx' + 2fy' +c,

where x', y' are the co-ordinates of the centre. The calculation of this may be facilitated by putting e' into the form

c' = [(ax' + hy' + g) x' + (hx' + by' + f') y'}+ gx' +ƒy' + c.

The first two sets of terms are rendered = 0 by the co-ordinates of the centre, and the last (Art. 140)

153. If the numerator of this fraction were = 0, the transformed equation would be reduced to the form

ax2+2hxy+by2 = 0,

and would, therefore (Art. 73), represent two real or imaginary right lines, according as ab-h2 is negative or positive. Hence,

* It is evident in like manner that the result of substituting x'y', the co-ordinates of the centre, in the equation of the polar of any point x"y", viz.

(ax' + hy' + g) x" + (hx' + by' +ƒ) y" + gx' +fy' + c,

is the same as the result of substituting x'y' in the equation of the curve. For the first two sets of terms vanish in both cases.