CHAPTER XII.

THE HYPERBOLA CONTINUED.

Diameters.

236. To find the length of a line drawn from any point in a given direction to meet an hyperbola.

Let x', y' be the co-ordinates of the point from which the line is drawn; x, y the co-ordinates of the point to which the line is drawn; the inclination of the line to the axis of x; r the length of the line; then (Art. 27)

x=x+r cos 0,

y=y'r sin 0............ (1).

If (x, y) be on the hyperbola these values may be substituted in the equation a2y2 — b2x2 = — a2b2; thus

--

a2 (y′ + r sin 0)2 — b2 (x' + r cos 0)2

- a2b2;

22 (a2 sin2 0 — b2 cos2 ) + 2r (a2y' sin 0 — b3x' cos 0)

+ a2y” — b2x22 + a2b2 = 0.................................. (2).

From this quadratic two values of r can be found which are the lengths of the two lines that can be drawn from (x', y') in the given direction to the hyperbola.

237. To find the diameter of a given system of parallel chords in an hyperbola. (See definition in Art. 148.)

Let be the inclination of the chords to the transverse axis of the hyperbola; let x', y' be the co-ordinates of the middle point of any one of the chords; the equation which determines the lengths of the lines drawn from (x', y') to the curve is (Art. 236)

r2 (a2 sin2 0 — b2 cos2 ) + 2r (a2y' sin ◊ — b3x' cos 0)

+a2y12 — b2x2 + a2b2 = 0 .................

(1).

Since (x', y') is the middle point of the chord, the values of furnished by this equation must be equal in magnitude and opposite in sign; hence the coefficient of r must vanish; thus

a2y' sin - b'x' cos 0=0, or y'= or y'cot 0. x'................. (2).

a

Considering x and y' as variable this is the equation to a straight line passing through the origin, that is, through the centre of the hyperbola.

Hence every diameter passes through the centre.

Also every straight line passing through the centre is a diameter, that is, bisects some system of parallel chords. For by giving to a suitable value the equation (2) may be made to represent any line passing through the centre. If ' be the inclination to the axis of x of the diameter which bisects all the chords inclined at an angle 0, we have from (2)

238. If one diameter bisect all chords parallel to a second diameter, the second diameter will bisect all chords parallel to the first.

2

Let 0, and 0, be the respective inclinations of the two diameters to the transverse axis of the hyperbola. Since the first bisects all the chords parallel to the second, we have

And this is also the only condition that must hold in order that the second may bisect the chords parallel to the first. The definition in Art. 191 holds for the hyperbola.

239. Every straight line passing through the centre of an ellipse meets that ellipse; this is evident from the figure, or it may be proved analytically. But in the case of an hyperbola this proposition is not true, as we proceed to shew.

240. To find the points of intersection of an hyperbola with a straight line passing through its centre.

Let the equation to the straight line be

y = mx.

Substitute this value of y in the equation to the hyperbola

then we have for determining the abscissæ of the points of intersection the equation.

Hence the values of x are impossible if a2m2 is greater than b2.

Thus a line drawn through the centre of an hyperbola will not meet the curve if it makes with the transverse axis on

either side of it an angle greater than tan‍

-1

b

a

241. It is convenient for the sake of enunciating many properties of the hyperbola to introduce the following important definition.

DEF. The conjugate hyperbola is an hyperbola having for its transverse and conjugate axes the conjugate and transverse axes of the original hyperbola respectively.

242. To find the equation to the hyperbola conjugate to a given hyperbola.

Let AA', BB' be the transverse and conjugate axes respectively of the given hyperbola; then BB is the transverse axis of the conjugate hyperbola, and AA' is its conjugate axis. Let P be a point in the given hyperbola, Q a point in the conjugate hyperbola. Draw PM, QN perpendicular to

CX, CY respectively. The equation to the given hyperbola

since is a point on an hyperbola having CB, CA for its semi-transverse and semi-conjugate axes respectively. Thus if x, y denote the co-ordinates of Q,

a3
x2 =
(y2 - b3).
b2

This, therefore, is the equation to the conjugate hyperbola; we observe that it may be deduced from the equation to the given hyperbola by writing - a2 for a2 and — ba for b2.

The foci of the conjugate hyperbola will be on the line BCB' at a distance from CAB (Art. 216); that is, at the same distance from C as S and H.

243. Every straight line drawn through the centre of an hyperbola meets the hyperbola or the conjugate hyperbola, except the two lines inclined to the transverse axis of the hyperbola

To find the abscissæ of the points of intersection of (1) with the given hyperbola, we have, as in Art. 240, the equation

a2b
b2 - a2m2

(2).

Similarly to find the points of intersection of (1) with the conjugate hyperbola, we have the equation

(3).

If m2 be less than, (2) gives possible values, and (3)

b2

impossible values for x; if m2 be greater than (2) gives

a2,
α

impossible values, and (3) possible values for x; if m2 =

Thus the two lines that can b

(2) and (3) make x infinite.
be drawn at an inclination tan

-1

a

Ito the transverse axis of

the given hyperbola meet neither curve; and every other line meets one of the curves.

244. Of two conjugate diameters one meets the original hyperbola, and the other the conjugate hyperbola.

Let the equations to the two diameters be