As increases from 0 to

π

cos e is negative when is greater than and r continues to

2

increase. Let a be such an angle that 1+e cos a = 0, that is, 1 then the nearer approaches to a the greater r

e

becomes, and by taking near enough to a, we may make r as great as we please. Thus as increases from 0 to a that portion of the curve is traced out which begins at A and passes on through P to an indefinite distance from the origin.

When 0 is greater than a, r is negative, and is at first indefinitely great and diminishes as increases from a to π. Sincer is negative we measure it in the direction opposite to that we should use if it were positive. Thus as increases from a to π that portion of the curve is traced out which begins at an indefinite distance from C in the lower left-hand quadrant, and passes on through Q to A'. HA' is found by putting in (1); then r becomes - a 0: =π -a (e+1), therefore

HA' is in length = a (e+1).

As increases from π to 2π-α, r continues negative and numerically increases, and may be made as great as we please by taking sufficiently near to 2π-a. Thus the branch of the curve is traced out which begins at A' and passes on through Q' to an indefinite distance.

α.

As increases from 2π-a to 2π, r is again positive, and is at first indefinitely great and then diminishes. Thus the portion of the curve is traced out which begins at an indefinitely great distance from C in the lower right-hand quadrant and passes on through P' to A.

The asymptotes CL and CL' are inclined to the transverse

axis at an angle of which the tangent is

=

α

√(a2 + b2)

b

a

1

hence cos LCA

and cos LCA'= ; that is, LCA' = a.

e

Thus as approaches the value a the radius vector approaches 0 to a position parallel to CL. Similarly as @ approaches the value 2π-α the radius vector approaches to a position parallel to CL'.

266. As in Art. 205 it may be shewn that the polar equation to a chord subtending at the focus an angle 2ẞ is

a-B and a+B being respectively the vectorial angles of the lines which join the focus to the ends of the chord, and 7 the semi-latus rectum.

Hence the polar equation to the tangent is

267. The polar equation to the hyperbola, the centre being the pole, is (Art. 206)

r2 (a sin-b2 cos20) = — a2b2.

Arts. 207, 208 are applicable to the Hyperbola.

Equilateral or Rectangular Hyperbola.

268. If in the equation to the ellipse ay2+ b2x2= a*V3, we suppose b=a, we obtain x+y=a', which is the equation to a circle; so that the circle may be considered a particular case of the ellipse. If in the equation to the hyperbola a3y3 — b2x2 = — a2b2 we suppose b=a, we have y-x=— a3. We thus obtain an hyperbola which is called the equilateral hyperbola from the equality of the axes. Since the angle between the asymptotes, which = 2 tan¬ becomes a right angle when ba, the equilateral hyperbola is also called the rectangular hyperbola.

-1

b

The peculiar properties of the rectangular hyperbola can be deduced from those of the ordinary hyperbola by making b=

= a.

Thus since b2=a* (e2 - 1) we have e3 − 1 = 1, ... e = √2. The equation to the tangent is (Art. 220)

yy' — xx' =— a2.

The equation to the conjugate hyperbola is, by Art. 242,

y3 - x2 = a2.

Thus the conjugate hyperbola is the same curve as the original hyperbola, though differently situated.

By Art. 248, CP= CD, and therefore by Art. 259, CP and CD are equally inclined to the asymptotes.

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.

a sin a

3. If a straight line be drawn from the focus of an hyperbola the part intercepted between the curve and the asymptote where ✪ and a are the angles made respectively by the straight line and asymptote with the axis.

sin a+ sin '

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 and 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 dif ference 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.

T. C. S.

15

7. A line is drawn parallel to the axis of y meeting the x2 y2

hyperbola =1, and its conjugate, in points P, Q; shew

a b2

that the normals at P and Q intersect each other on the axis of x. Shew also that the tangents at P and Q intersect on the curve whose equation is y* (a2y2 — b2x2) = 4box2.

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 equation 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 tangents to a parabola include an angle a; shew that the locus of their point of intersection is an hyperbola with the same focus and directrix.

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

11. The ratio of the sines of the angles made by a diameter of an hyperbola with the asymptotes is equal to the ratio of the sines of the angles made by the conjugate diameter.

12. With two conjugate diameters of an ellipse as asymptotes a pair of conjugate hyperbolas is constructed; prove that if one hyperbola touch the ellipse the other will do so likewise; prove also that the diameters drawn through the points of contact are conjugate to each other.

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

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

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

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