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 + 2ƒy+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 y2 = 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 y=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 y3 = 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 + 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 − k2 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)

whence

a (g−ak) + B (fƒ — ßk) = 0,

k =

ag + Bf
a2 + B* •

2

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 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)

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

whence

yY=ax+ By, yX=ßx-ay,

yx=aY+BX, yy=BY-aX.

Making these substitutions in the equation of the curve it becomes y3 Y2 + 2 (gß − 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 + 2ƒ'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 + 29′x + 2 (b'y′ +f') 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 y=px. The actual values of the coordinates of the new

When the equation of a parabola is reduced to the form y3 = 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

(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, we 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

which becomes F2 3X when transformed to parallel axes through (— §, − 1).

=

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 by2 + 2ƒ'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

But if we substitute for a, h, b, respectively, a", aß, B2, the lefthand side of the equation vanishes, and the right-hand side becomes (fa-gB). Writing the condition fa=gẞ in either of the forms fa2 = gaß, faß = gẞ", we see that the general equation of the second degree represents two parallel right lines when hab, 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 cosa) x (a — ẞ cos w) 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 cosa) x-(a-ẞ cos w) y; whence

yx sin∞ = (a - ẞ cos w) Y+ BX sin w;

ух

yy sin w = (B-a cos ∞) Y-aX sin w.

Making these substitutions, the equation becomes

y3 Y2 + 2 sin2 w (gß – fa) X

W

+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 y=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 x 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.

V F

P

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+l) meets the parabola (y2 = px), we obtain the quadratic

y2 - pky - pl = 0,

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 the point x= ; and although this value increases as m increases, yet it will never become infinite as long as m is finite.

m2

Ρ

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