Ex. 2. Find the coordinates of the intersection of the tangents at x'y', x'y'.
Solve for x and y from

200. To express the quantity k in terms of the lengths of the axes of the curve.

Since the axis bisects the angle between the asymptotes, the coordinates of its vertex are found, by putting x=y in the equation xy=k', to be x=y=k.

Hence, if be the angle between the axis and the asymptote

a = 2k cos 8,

(since a is the base of an isosceles triangle whose sides = k and base angle), but (Art. 165)

And the equation of the curve, referred to its asymptotes, is

201. The perpendicular from the focus on the asymptote is equal to the conjugate semi-axis b.

For it is CF sine, but CF = √(a2 + b2), and sin0 =

b

√(a2 + b2)

This might also have been deduced as a particular case of the property, that the product of the perpendiculars from the foci on any tangent is constant, and = b. For the asymptote may be considered as a tangent, whose point of contact is at an infinite distance (Art. 154), and the perpendiculars from the foci on it are evidently equal to each other, and on opposite sides of it.

202. The distance of the focus from any point on the curve is equal to the length of a line drawn through the point parallel to an asymptote to meet the directrix.

CC

For the distance from the focus is e times the distance from the directrix (Art. 186), and the distance from the directrix is to

the length of the parallel line as cos

1

=

e

167)

Art. 167 is to 1.

Hence has been derived a method of describing the hyperbola

by continued motion. A ruler ABR, bent D
at B, slides along the fixed line DD; a
thread of a length = RB is fastened at the
two points R and F, while a ring at P keeps
the thread always stretched; then, as the
ruler is moved along, the point P will de-
scribe an hyperbola, of which F is a focus,
DD a directrix, and BR parallel to an
asymptote, since PF must always = PB.

B

A

F

R

P

CHAPTER XII.

THE PARABOLA.

REDUCTION OF THE EQUATION.

203. THE equation of the second degree (Art. 137) will represent a parabola, when the first three terms form a perfect square, or when the equation is of the form

(ax + By)2+2gx+2fy+c=0.

We saw (Art. 140) that we could not transform this equation so as to make the coefficients of x and y both to vanish. The form of the equation, however, points at once to another method of simplifying it. We know (Art. 34) that the quantities ax+ By, 2gx + 2fy+c, are respectively proportional to the lengths of perpendiculars let fall from the point (xy) on the right lines, whose equations are

ax+By=0, 2gx + 2fy + c = 0.

Hence, the equation of the parabola asserts that the square of the perpendicular from any point of the curve on the first of these lines is proportional to the perpendicular from the same point on the second line. Now if we transform our equation, making these two lines the new axes of coordinates, then since the new x and y are proportional to the perpendiculars from any point on the new axes, the transformed equation must be of the form y=px.

The new origin is evidently a point on the curve; and since for every value of x we have two equal and opposite values of y, our new axis of x will be a diameter whose ordinates are parallel to the new axis of y. But the ordinate drawn at the extremity of any diameter touches the curve (Art. 145); therefore the new axis of y is a tangent at the origin. Hence the line ax + By is the diameter passing through the origin, and 2gx + 2fy + c is the tangent at the point where this diameter meets the curve. And the equation of the curve referred to a diameter and tangent at its extremity, as axes, is of the form y2=px.

204. The new axes to which we were led in the last article are in general not rectangular. We shall now show that it is possible to transform the equation to the form y=px, the new axes being rectangular. If we introduce the arbitrary constant k, it is easy to verify that the equation of the parabola may be written in the form

(ax+ By + k)*+2(g−ak) x + 2 (ƒ − ẞk) y + c − k2 = 0.

Hence, as in the last article, ax + By + k is a diameter, 2 (g− ak) x + 2 (ƒ − ẞk) y + c − k is the tangent at its extremity, and if we take these lines as axes, the transformed equation is of the form y=px. Now the condition that these new axes should be perpendicular is (Art. 25)

Since we get a simple equation for k, we see that there is one diameter whose ordinates cut it perpendicularly, and this diameter is called the axis of the curve.

205. We might also have reduced the equation to the form y=px by direct transformation of coordinates. In Chap. XI. we reduced the general equation by first transforming to parallel axes through a new origin, and then turning round the axes so as to make the coefficient of xy vanish. We might equally well have performed this transformation in the opposite order; and in the case of the parabola this is more convenient, since we cannot, by transformation to a new origin, make the coefficients of x and y both vanish.

We take for our new axes the line ax + By, and the line perpendicular to it ẞx - ay. Then since the new X and Y are to denote the lengths of perpendiculars from any point on the new axes, we have (Art. 34)

If for shortness we write a +B=y, the formulæ of transformation become

whence

yY= = ax + By, yx=αY+BX,

yX= ßx — ay,

y = BY- aX.

Making these substitutions in the equation of the curve it becomes

2

y3 Y2 + 2 (gB − fa) X + 2 (ga +ƒB) Y+yc= 0.

Thus, by turning round the axes, we have reduced the equation to the form

b'y2+2g'x+2f'y + c' = 0.

If we change now to parallel axes through any new origin x'y', substituting x + x, y+y" for x and y, the equation becomes

b'y2 + 2g'x + 2 (b'y' +f') y + b'y22 + 2g'x' + 2ƒ'y' + c′ = 0.

The coefficient of x is thus unaltered by transformation, and therefore cannot in this way be made to vanish. But we can evidently determine x and y', so that the coefficients of y and the absolute term may vanish, and the equation thus be reduced to y=px. The actual values of the coordinates of the new origin are y'

f'
b

=

x' = ƒ" - b'c'
2g'b'

;

and p is evidently

2g'

or

in terms of the original coefficients

(a2 + B2)*

When the equation of a parabola is reduced to the form y2 = px, the quantity p is called the parameter of the diameter, which is the axis of x; and if the axes be rectangular, p is called the principal parameter (see Art. 194).

Ex. 1. Find the principal parameter of the parabola

9x2+24xy+16y2 + 22x + 46y + 9 = 0.

First, if we proceed as in Art. 204, we determine k = 5. The equation may then be written

Now if the distances of any point from 3x + 4y + 5 and 4x-3y+ 8 be Y and X, we have

The process of Art. 205 is first to transform to the lines 3x+4y, 4x-3y as axes, when the equation becomes

which becomes Y2 = 3X when transformed to parallel axes through (- §, − 1).

Ex. 2. Find the parameter of the parabola

This value may also be deduced directly by the help of the following theorem, which will be proved afterwards :-"The focus of a parabola is the foot of a perpendi