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 ( f − 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 − k2 is the tangent at its extremity, and if we take these lines as axes, the transformed equation is of the form y2=px. Now the condition that these new axes should be perpendicular is (Art. 25)

whence

a (g − ale) + B (ƒ— ßk) = 0,

k = ag + Bf
a2+B2

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 co-ordinates. 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 Bx-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)

Making these substitutions in the equation of the curve, it becomes

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

Thus, by turning round the axes, we have reduced the equation to the form b'y* + 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' +ƒ') y + b'y'2 + 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 y2=px. The actual values of the co-ordinates of the new

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. J

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

(3x+4y+ 5)2 = 2 (4x − 3y + 8).

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

5Y=3x+4y+5, 5X=4x-3y + 8,

and the equation may be written Y2 = 3X.

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

25Y2+50Y - 10X + = = 0,

or

25 (Y+1)2 = 10X + 16,

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

Ex. 2. Find the parameter of the parabola

y2

22 - 2xy + 1/2 - 20 - 21 +
2x 2y + 1 = 0.

a2 ab b2 α

4a2b2

Ans.

(a2 + b2);

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

cular let fall from the intersection of two tangents which cut at right angles on their chord of contact ;" and "The parameter of a conic is found by dividing four times the rectangle under the segments of a focal chord, by the length of that chord" (Art. 193 Ex. 1).

Ex. 3. If a and b be the lengths of two tangents to a parabola which intersect at, right angles, and m one quarter of the parameter, prove

206. If in the original equation gẞ=fa, the coefficient of x vanishes in the equation transformed as in the last article; and that equation b'y2+2f'y + c' = 0, being equivalent to one of the form b' (y — λ) (y — μ) = 0,

represents two real, coincident, or imaginary lines parallel to the new axis of x.

We can verify that in this case the general condition that the equation should represent right lines is fulfilled. For this condition may be written

c (ab — h3) = af3 — 2hfg +bg3.

But if we substitute for a, h, b, respectively, a3, aß, ß2, the lefthand side of the equation vanishes, and the right-hand side becomes (fa-gẞ)". Writing the condition fa=gẞ in either of the forms fa" = gaß, faß=gB", we see that the general equation of the second degree represents two parallel right lines when h2 = ab, and also either af=hg, or fh=bg.

*207. If the original axes were oblique, the equation is still reduced, as in Art. 205, by taking for our new axes the line ax+By, and the line perpendicular to it, whose equation is (Art. 26) (B − a cos w) x − (a - ẞ cosw) y = 0.

And if we write y2 = a2 + ß2 – 2aß cosa, the formulæ of transformation become, by Art. 34,

yY= (ax+ẞy) sin w, yX=(B- a cos w) x — (a — ß cosw) y; whence yx sin w = (a - ẞ cosw) Y+BX sino;

yy sin w = (B-a cosa) Y- aX sinw.

Making these substitutions, the equation becomes

y3 Y2+2 sin3w (gẞ – fa) X

+2 sin∞ {g (a-ẞ cos w) +ƒ(B-a cos w)} Y+yc sin' w = 0.

And the transformation to parallel axes proceeds as in Art. 205. The principal parameter is

208. From the equation y2=px we can at once perceive the figure of the curve. It must be symmetrical on both sides of the axis of x, since every value for x gives two equal and opposite for y. None of it can lie on the negative side of the origin, since if we make a negative, y will be imaginary, and as we give increasing positive values to x, we obtain increasing values for y. Hence the figure of the curve is that here represented.

VF

M

Although the parabola resembles the hyperbola in having infinite branches, yet there is an important difference between the nature of the infinite branches of the two curves. Those of the hyperbola, we saw, tend ultimately to coincide with two diverging right lines; but this is not true for the parabola, since, if we seek the points where any right line (x=ky+1) meets the parabola (y2 = px), we obtain the quadratic

whose roots can never be infinite as long as k and 7 are finite. There is no finite right line which meets the parabola in two coincident points at infinity; for any diameter (y=m), which meets the curve once at infinity (Art. 142), meets it once also in

2

m2

the point x= ; and although this value increases as m in

creases, yet it will never become infinite as long as m is finite.

209. The figure of the parabola may be more clearly conceived from the following theorem: If we suppose one vertex

and focus of an ellipse given, while its axis major increases without limit, the curve will ultimately become a parabola.

The equation of the ellipse, referred to its vertex, T is (Art. 194)

F

x2.

Р

F

We wish to express b in terms of the distance VF(=m),

whence b2 = 2am — m3, and the equation becomes

√(a - b) (Art. 182),

Now, if we suppose a to become infinite, all but the first term of the right-hand side of the equation will vanish, and the equation becomes

the equation of a parabola.

y2 = 4mx,

A parabola may also be considered as an ellipse whose eccen

tricity is equal to 1. For e=1

2

a2

which is the coefficient of x in the preceding equation, vanished as we supposed a increased according to the prescribed conditions; hence e2 becomes finally = 1.

THE TANGENT.

210. The equation of the chord joining two points on the curve is (Art. 86) (y—y') (y — y′′) = y2 — px,

And if we make y"=y', and for y" write its equal px', we have the equation of the tangent

2y'y = p(x+x').

If in this equation we put y = 0, we get x=-x': hence TM (see fig. next page) (which is called the Subtangent) is bisected at the vertex.

These results hold equally if the axes of co-ordinates are oblique; that is to say, if the axes are any diameter and the tangent at its vertex, in which case we saw (Art. 203) that the equation of the parabola is still of the form y=p'x.