EXAMPLES.

1. The radius of a circle which touches an hyperbola and its asymptotes is equal to that part of the latus rectum which. is intercepted between the curve and asymptote.

2. A line drawn through one of the vertices of an hyperbola and terminated by two lines drawn through the other vertex parallel to the asymptotes will be bisected at the other point where it cuts the hyperbola.

3. If a straight line be drawn from the focus of an hyperbola the part intercepted between the curve and the asymptote

=

a sin a

sin a + sin @ where and a are the angles made respectively by the straight line and asymptote with the axis.

4. PQ is one of a series of chords inclined at a constant angle to the diameter AB of a circle, find the locus of the point of intersection of AP, BQ.

5. Pis a point in a branch of an hyperbola, P' is a point in a branch of its conjugate, CP, CP', being conjugate semidiameters. If S, S', be the interior foci of the two branches, prove that the difference of SP and S'P' is equal to the difference of AC and BC.

6. If x, y, be co-ordinates of any point of an hyperbola, shew that we may assume x = a sec 0, y = b tan 0.

x2 y2
b2

a2

7. A line is drawn parallel to the axis of y meeting the hyperbola = 1, and its conjugate, in points P, Q; shew that the normals at P and Q intersect each other on the axis Shew also that the tangents at P and Q intersect on the curve whose equation is y* (a'y' — b2x2) = 46°x2.

of x.

8.

Tangents to an hyperbola are drawn from any point in one of the branches of the conjugate; shew that the chord of contact will touch the other branch of the conjugate.

Find the equations to the radii from the centre to the points of contact of the two tangents, and if these radii are perpendicular to one another, shew that the co-ordinates of the point from which the tangents are drawn are

9. Two lines are drawn through the focus of an ellipse including a constant angle; tangents are drawn to the ellipse at the points where the lines meet the ellipse; find the locus of the intersection of the tangents.

10. Under what limitation is the proposition in Example 30 of Chapter x. true for the hyperbola?

CHAPTER XIII.

GENERAL EQUATION OF THE SECOND DEGREE.

269. WE shall now shew that every locus represented by an equation of the second degree is one of those which we have already discussed, that is, is one of the following; a point, a straight line, two straight lines, a circle, a parabola, an ellipse, or an hyperbola.

The general equation of the second degree may be written

ax2 + bxy + cy2+ dx + ey +ƒ= 0;

we shall suppose the axes rectangular; if the axes were oblique we might transform the equation to one referred to rectangular axes, and as such a transformation cannot affect the degree of the equation (Art. 87), the transformed equation will still be of the form given above.

If the curve passes through the origin f=0; if the curve does not pass through the origin f is not =0, we may therefore divide by ƒ and thus the equation will take the form

a2x2 + b'xy + c'y2 + d'x + e'y +1 = 0.

270. We shall first investigate the possibility of removing from the equation the terms involving the first powers of the variables.

Transfer the origin of co-ordinates to the point (h, k) by putting

x=x+h, y=y' + k,

and substituting these values of x and y in the equation

х

ax2 + bxy + cy2+ dx+ey+f=0... (1);

thus we obtain

2

ax2 + bx'y' + cy'2 + (2ah + bk + d) x' + (2ck + bh + e) y'

where

+ƒ' =0.........(2),

f' = ah2 + bhk + ck2 + dh + ek +ƒ....................... (3). Now, if possible, let such values be assigned to h and k as to make the coefficients of x' and y' vanish; that is, let

=

2ah + bk + d = 0, and 2ck+bh + e = 0;

It will therefore be possible to assign suitable values to h and k, provided b2-4ac be not = 0.

We shall see that the loci represented by the general equation of the second degree may be separated into two classes, those which have a centre, and those which in general have not a centre, and that in the former case b2-4ac is not zero, and in the latter case it is zero. We shall first consider the case in which b2-4ac is not zero, and consequently the values found above for h and k are finite.

Equation (2) thus becomes

ax'2 + bx'y' + cy'2 +ƒ” = 0 ..........................

(4).

Now if (4) is satisfied by any values x1, y1, of the variables, it is also satisfied by the values - x1, Hence the new origin of co-ordinates is the centre of the locus represented by (1).

Thus if b2-4ac be not = 0, the locus represented by (1) has a centre, and its co-ordinates are h and k, the values of which are given above.

The value of f' may be found by substituting the values of h and k in (3); the process may be facilitated thus; we have

2ah+bk + d = 0,

2ck+bh + e = 0.

Multiply the first of these equations by h, and the second by k, and add; thus

271. We may suppress the accents on the variables in (4) of the preceding article and write it

ax2 + bxy + cy2+ƒ' = 0...............(5).

This equation we shall further simplify by changing the directions of the axes. (Art. 81.)

Put

x = x' cos - y' sin 0,

y=x' sin+y' cos 0,

and substitute in (5); thus

x2(a cos20+ c sin20+b sin cos 0)

+y'2 (a sin2 0 + c cos2 0 -b sin cos 0)

+ x'y' {2 (c − a) sin ✪ cos 0) + b (cos20 — sin20)} +ƒ' = 0.....(6).

or

Equate the coefficient of x'y' to zero; thus

2 (c − a) sin ✪ cos 0 + b (cos2 0 — sin2 0) = 0,

Since can always be found so as to satisfy (7), the term involving x'y' can be removed from (6), and the equation becomes