negative; that is to say, that one of the conics is an ellipse and the other a hyperbola. Considering then b' as containing implicitly a negative sign, the values we have obtained for the coordinates may be written symmetrically

194 (c). From the second term in either of the equations we get an expression for the square of the radius vector to the point P, viz.

x" + y2 = a" + a""2 — c2 = a"2 + b′′ = b'" + a′′2.

This also may be got by adding the expressions for x" and y' just found, since

The square of the semi-diameter of the ellipse conjugate to CP is given by the equation 6" = aˆ* + b′′a − (a3 +b''*), and is therefore b"-b""" or a" - a"".

If p' be the perpendicular on the tangent to the ellipse at P, we have ẞp' a'b', and therefore

In like manner if p" be the perpendicular on the tangent to the hyperbola we have

The reader will observe the symmetry which exists between these values for p", p"", and the values already found for x", y'. If the two tangents at P be taken as axes of coordinates, p', p" are the coordinates of the centre C. The analogy then between the values for p', p" and those for x', y' may be stated as follows: With the point P as centre, two confocal conics may be described having the tangents at P as axes, and intersecting in C. The axes of the new system are a', a"; b', b′′; and the tangents at C to the new system are the axes of the old system.

194 (d). Returning to the quadratic of 194 (a), if x”, λ be the roots, we have λ'2λ””2 = a2b2 — b2x22 — a2y'3.

112

Now if x'y' be a point external to + =1, we have λ"a" - a",

=

y

b2

X""a" - a2; and it will be observed that "" is essentially negative, since the axis

than that of any ellipse.

of any hyperbola of the system is less

Thus we have

The expression given (Ex. 3, Art. 169) for the angle between the tangents to an ellipse from an external point may be thrown into the form

We have seen (Art. 189) that the tangents PT, Pt are equally inclined to the tangent to the confocal ellipse at P, or, in other words, that that tangent is the external bisector of the angle TPt. If then that tangent make an angle with PT, will be the complement of 14, and we have

--

COR. 2. If on the tangents PT, Pt be taken from P portions, equal respectively to the focal distances PF, PF', the length of the line joining their extremities will be 2a. For if we consider the triangle whose sides are a'+a”, a′ — a′′ (see Art. 194a) and 2a, and apply the ordinary trigonometric formula (sa) (8-b) tan2C = we find for the angle between the first " 8 (8-c) two lines the same value as that just found for p.

COR. 3. If from a point P tangents be drawn to two fixed confocal ellipses, the ratio (sin : sin') of the sines of the

190

angles which these tangents make with the tangent to the confocal ellipse passing through P will be constant while P moves on that ellipse. For if a and A be the semi-axes of the interior ellipses, we have, from what has been just proved,

an expression not involving a", and therefore the same for every point on the ellipse a'.

THE ASYMPTOTES.

195. We have hitherto discussed properties common to the ellipse and the hyperbola. There is, however, one class of properties of the hyperbola which have none corresponding to them in the ellipse, those, namely, depending on the asymptotes, which in the ellipse are imaginary.

We saw that the equation of the asymptotes was always obtained by putting the terms containing the highest powers of the variables = 0, the centre being the origin. Thus the equation of the curve, referred to any pair of conjugate diameters, being

Hence the asymptotes are parallel to the diagonals of the parallelogram, whose adjacent sides are any pair of conjugate semidiameters. For, the equation of

is parallel to the other(see Art.167).

T

Hence, given any two conjugate diameters, we can find the asymptotes; or, given the asymptotes, we can find the diameter conjugate to any given one; for if we draw 40 parallel to one asymptote, to meet the other, and produce it till OB= A0, we find B, the extremity of the conjugate diameter.

196. The portion of any tangent intercepted by the asymptotes is bisected at the curve, and is equal to the conjugate diameter.

This appears at once from the last Article, where we have proved AT=b = AT; or directly, taking for axes the diameter through the point and its conjugate, the equation of the asymp

Hence, if we take x=a', we have y=+b; but the tangent at A being parallel to the conjugate diameter, this value of the ordinate is the intercept on the tangent.

197. If any line cut a hyperbola, the portions DE, FG, intercepted between the curve and its asymptotes, are equal.

For, if we take for axes a diameter parallel to DG and its conjugate, it appears from the last Article that the portion DG is bisected by the diameter; so is also the portion EF; hence DE=FG.

T

The lengths of these lines can immediately be found, for,

from the equation of the asymptotes (7

b

y (=DM=MG)=±√x.

a

Again, from the equation of the curve

we have

Hence

and

a

y (= EM=FM) = ±b′ √(-1).

= b'

=0),

we have

DE (FG) - V-√(-1)},

=

198. From these equations it at once follows that the rectangle DE.DF is constant, and b. Hence, the greater DF is, the smaller will DE be. Now, the further from the centre we draw DF the greater will it be, and it is evident from the value

given in the last article, that by taking a sufficiently large, we can make DF greater than any assigned quantity. Hence, the further from the centre we draw any line, the less will be the intercept between the curve and its asymptote, and by increasing the distance from the centre, we can make this intercept less than any assigned quantity.

199. If the asymptotes be taken for axes, the coefficients g and of the general equation vanish, since the origin is the centre; and the coefficients a and b vanish, since the axes meet the curve at infinity (Art. 138, Ex. 4); hence the equation reduces to the form

The geometrical meaning of this equation evidently is, that the area of the parallelogram formed by the coordinates is constant. The equation being given in the form ay=k, the equation of any chord is (Art. 86),

or

=

(x-x') (y-y') = xy — k3,

x'y + y′′x = k2 + x'y'".

Making xx" and y'=y", we find the equation of the tangent x'y + y'x = 2k3,

From this form it appears that the intercepts made on the asymptotes by any tangent = 2x and 2y'; their rectangle is, therefore, 4k2. Hence, the triangle which any tangent forms with the asymptotes has a constant area, and is equal to double the area of the parallelogram formed by the coordinates.

Ex. 1. If two fixed points (x'y', x'y') on a hyperbola be joined to any variable point on the curve (x"y""), the portion which the joining lines intercept on either asymptote is constant.

The equation of one of the joining lines being

x""y + y'x = y'x'"' + k2,

the intercept made by it from the origin on the axis of x is found, by making y = 0, to be x'"+x'. Similarly the intercept from the origin made by the other joining line is "'+x", and the difference between these two (x' - x') is independent of the position of the point a""y"".