The corresponding equation for the circle is (a being the semi-diameter)

y2=2ax-x2.

If A and B have unlike signs we may obtain from (IV) in a similar manner, for the hyperbola (see the note),

But these equations for the ellipse and hyperbola may be obtained more readily on substituting a-x for x in the equation

b2
y2 = 2 (a2x2),

a2

and a +x for x in the equation

b2 a2

y2= (x2 — a2).

b2

59. Since, in the ellipse and hyperbola represents it

B

A

a2,

of an asymptote referred to two conjugate diameters becomes, omitting the sub-accents,

which, when the conjugate semi-diameters are the major and minor axes of the curve, is

60. COR. The double sign prefixed to the values of y indi

In a similar manner, for the hyperbola, making the above value of

B negative (A and в having unlike signs), may be obtained the equation

A

cates that for every value of x there are two equal values of that ordinate.

T

Thus, let ox, OY be co-ordinate axes coincident with two conjugate diameters of an hyperbola; then, for every value of x (OR) there will be two equal values (RT and RT) of y, the points T and T' being in the asymptotes oz, oz'. But for every value of x there are (Art. 57. a.) two equal values of the corresponding ordinates (RP, RP) of the hyperbola; therefore, taking equals from equals, PT=

P' T'.

T/

(P/

R

Χ

Thus when any chord in an hyperbola is produced to meet the asymptotes, the segments between the curve and the asymptotes are equal to one another. It follows also, VAV' being a tangent at A, that AV AV' and that each is equal to b.

PROPOSITION III.

61. To find the equation for an hyperbola with respect to the asymptotes in terms of the transverse and conjugate axes.

From Art. 53. a. we have, when the co-ordinate axes are coincident with the asymptotes

ox, or of an hyperbola,

Y

P/

X

gate axis; and let the asymptote Oy be referred to these axes; then the equation of the asymptote will be

consequently, when r,=t (=OA), y,= ±c (=AV or AV' drawn through a at right angles to OA, to meet oY and ox). Now, in the triangles VOA, AOV', AV=AV', OA is common, and the angles OAV, OAV' are right angles; therefore the angle Vov' between the asymptotes is bisected by OA; also YOX' is bisected by oz. Next, AK being drawn parallel to ox and AH to Oy, it may be proved that OK, KV, KA, OH, HV′, HA are equal to one another; for the angle vov' being bisected by

OA and the alternate angles at A being equal; the angles AOK, OAK (=AOH) are equal to one another, and consequently OK KA; in like manner OHHA, and the triangles OKA, OHA being equal, OKOH, &c.; also OAV being a right angle, VAK and AVK are each equal to the complement of AOK; therefore KV=KA, and similarly HV′=HO=HA. But ov2 (=0A2+AV2) = t2+c2, and OK2=40v2; therefore OK2, OH2 and HA2 are each equal to (t+c2). Now OH and HA are particular values of x and y, the co-ordinates of the curve referred to the asymptotes; therefore, xy (=F') being constant,

xy=} (t2+c2).

If the second member be represented by m2, we have

xy=m2.

62. COR. If any number of parallelograms, as OP, OP', be formed by drawing lines, as PQ, P'Q', parallel to the asymptote Oy, those parallelograms will manifestly be equivalent to one another; for the angle Yox being common to all, let it be represented by w; then OH and HA being each represented by m, oq or oq by x, and QP or Q'P' by y, on multiplying both members of the above equation by sin. w we have

xy sin. wm2 sin. w:

and (Pl. Trigon. Art. 75. a.) these members denote, respectively, the areas of the parallelograms OP, or OP′ and KH: thus the equivalence of the parallelograms is proved.

a. Since x and y represent the co-ordinates of any point, as P, P', &c., it is evident, from the equation xy=m2, that among the co-ordinates of points in the hyperbola, with respect to the asymptotes, there exist the following proportions:

OH OQ: PQ: OK, whence

OH OQ P'Q' : OK, &c.;

OH. OK=0Q. PQ=0Q'. P'Q', &c.,

or OH, OQ, OQ', &c. are reciprocally as OK, PQ, P'q', &c.

PROPOSITION IV.

63. To find the equation for a parabola in terms of an abscissa or segment of any diameter and the corresponding ordinate.

The equation for a parabola, when one of the co-ordinate axes coincides with a diameter and the other is parallel to

M

one of the ordinates of that diameter, or coincides with a tangent to the curve at the extremity of the diameter (the origin of the co-ordinates being then at that extremity of the diameter), is (Arts. 46. a. and 49.),

Ay2+Ex=0.

Let ox, OY be the co-ordinate axes, and MON a portion of the parabolic curve. Let

any segment OR, of ox, be represented by

a; RP, the corresponding ordinate, by b;

Y

R

X

that is, when x=a, let y=b. Then the equation becomes, (E being considered as negative),

If the origin of the co-ordinates were at A in XO produced, an equation for the parabola, in the like terms, might be obtained from the equation

Ay2+ Ex=F;

or, more simply (representing ao by m, and AR being a), by substituting x-m for x in the equation above for y2; which gives, a and b having the same values as before,

64. The equations for an ellipse, an hyperbola, and a parabola, may be converted into proportions, in which form the relations between the co-ordinates x and y are frequently stated.

Thus,
From (a) Art. 56. we have

or

or again, from (a), Art. 58. From (a) art. 57. we have

or

or again, from (b), Art. 58. and from (a) Art. 63. we have

In the six first proportions, or those which appertain to the ellipse and hyperbola, the terms a2-2 and 2 ax-x2, x2-a2 and 2 ax+2 denote the products of the abscissæ or segments, between the place of the ordinate y and the two extremities of the diameter. Of these curves, therefore, it

may be stated that the products of the abscissæ of any diameter, are to one another as the squares of the corresponding ordinates. The last proportion, or that which appertains to a parabola, indicates that the abscisse of any diameter are to one another as the squares of the corresponding ordinates.

a. COR. to the proposition. If a tangent at the extremity of a diameter, as AB of a parabola, be cut in any points T and T' by diameters drawn through points, as P and P' in the curve, we shall have

PT PT AT2: AT/2.

T

T

For drawing ordinates PR, P'R' to the diameter AB, these will be parallel to AT', and the diameters being parallel to one another, the figures RT, R'T' will be parallelograms; hence AR, AR' will be respectively equal to TP, T'P' and RP, R'P' to AT, AT'. Now (as above) AR: AR': RP2: R' P'2; therefore, substituting equals for equals,

PT PT AT2: AT'2.

A

R

R

B

65. DEF. 11. A third proportional to any diameter and its conjugate, is called the parameter of that diameter.

Thus, in an ellipse or hyperbola, a being any semi-diameter and b one which is conjugate to it; also, p representing the parameter of the former diameter,

consequently

2a 2b 2b: p;

:

262 p= a

In a parabola, a, on any diameter, being an abscissa measured from the vertex or intersection of the diameter with the curve line, and b the corresponding ordinate, the proportion is

abb: p;

therefore the parameter of any diameter of a parabola is exb2 pressed by

The parameter of the transverse axis of an ellipse or hyperbola is sometimes called the latus rectum. Also a constant line appertaining to any curve, or a term in the equation for any curve, which, on being made to vary, adapts the equation to other curves of a like kind, is called a parameter.