249. The above definition corresponds to the equation y2 = b2 / a2 (2ax - x2) when the vertex is the origin.

250. AN.NA' is equal to the square on the ordinate QN of the auxiliary circle, and PN : QN = PC: AC.

251. Fig. 71 shows how the points can be determined when the constant ratio is less than unity. Thus, layoff CD = AC, the semi-major axis. Through E any point of A C draw DE and produce it to meet the auxiliary circle in Q. Draw B'E and produce it to meet the ordinate QN in P. Then is PN: QN = B'C: DC=BC: AC. The same process is applicable when the ratio is greater than unity. When points in one quadrant are found, corresponding points in other quadrants can be easily marked.

Fig. 71.

252. If P and P' are the extremities of two conjugate diameters of an ellipse and the ordinates MP and M'P meet the auxiliary circle in Q and Q', the angle QCQ' is a right angle.

Now take a rectangular piece of card or paper and mark on two adjacent edges beginning with the common corner lengths equal to the minor and major axes. By turning the card round C mark corresponding points on the outer and inner auxiliary circles. Let Q, R and Q', R' be the points in one position. Fold the ordinates QM and Q'M', and RP and R'P', perpendiculars to the ordinates. Then P and P' are points on the curve.

Fig. 72.

253. Points on the curve may also be easily determined by the application of the following property of the conic sections.

The focal distance of a point on a conic is equal to the length of the ordinate produced to meet the tangent at the end of the latus rectum.

254. Let A and A' be any two points. Draw AA' and produce the line both ways. From any point D in A'A produced draw DR perpendicular to AD. Take any point R in DR and draw RA and RA'. Fold AP perpendicular to AR, meeting RA' in P. For different positions of R in DR, the locus of P is an ellipse, of which AA' is the major axis.

Fig. 73.

Fold PN perpendicular to A A'. Now, because PN is parallel to RD,

PN:A'N=RD:A'D. Again, from the triangles, APN and DAR, PN:AN=AD: RD. ... PN2: AN.A'N=AD: A'D, a constant ratio, less than unity, and it is evident from the construction that N must lie between A and A'.

Section IV. - The Hyperbola

255. An hyperbola is the curve traced by a point which moves in a plane in such a manner that its distance from a given point is in a constant ratio of greater inequality to its distance from a given straight line.

256. The construction is the same as for the ellipse, but the position of the parts is different. As explained in § 119, X, A' lies on the left side of the directrix. Each directrix lies between A and A', and the foci lie without these points. The curve consists of two branches which are open on one side. The branches lie entirely within two vertical angles formed by two straight lines passing through the center which are called the asymptotes. These are tangents to the curve at infinity.

257. The hyperbola can be denned thus : If a point P move in such a manner that PN2 : AN • NA' is a constant ratio, PN being the distance of P from the line joining two fixed points A and A', and N not being between A and A', the locus of P is an hyperbola, of which AA' is the transverse axis.

This corresponds to the equation y2 = b2/a2 (2ax+xt), where the origin is at the right-hand vertex of the hyperbola.

Fig. 74 shows how points on the curve may be found by the application of this formula.

Let C be the center and A the vertex of the curve.

CB'=CB = b; CA' = CA = CA' = a.

Fold CD any line through C and make CD = CA. Fold DN perpendicular to CD. Fold NQ perpendicular to CA and make NQ = DN. Fold QA" cutting CA in S. Fold B'S cutting QN in P.

Fig. 74.

Then P is a point on the curve. For, since DN is tangent to the circle on the diameter A'A

DN2 = AN. (2CA + AN), or since QN=DN,

QN2 = x(2a + x).

QN / PN = A"C / BC.

Squaring, x(2a + x) / y2 = a2 / b2 , or y2 = b2 / a2 (2ax + x2).

If QN = b then N is the focus and CD is one of the asymptotes. If we complete the rectangle on AC and BC the asymptote is a diagonal of the rectangle.

258. The hyperbola can also be described by the property referred to in § 253.

259. An hyperbola is said to be equilateral when the transverse and conjugate axes are equal. Here a = b, and the equation becomes y2 = (2a + x)x. In this case the construction is simpler as the ordinate of the hyperbola is itself the geometric mean between AN and A'N, and is therefore equal to the tangent from N to the circle described on A A' as diameter.

260. The polar equation to the rectangular hyperbola, when the center is the origin and one of the axes the initial line, is r2 cos 2θ = a2 or r2 = a / cos 2θ. a.

Let OX, OY be the axes; divide the right angle YOX into a number of equal parts. Let XOA, AOB be two of the equal angles. Fold XB at right angles to OX. Produce BO and take OF= OX. Fold OG perpendicular to BF and find G in OG such that FGB is a right angle. Take OA = OG. Then A is a point on the curve.

Fig. 75.

Now, the angles XOA and AOB being each θ,

OB= a / cos 2 θ

And OA2=OG2=OB •OF = a / cos 2θ .a.

...r2cos2θ = a2.

261. The points of trisection of a series of conterminous circular arcs lie on branches of two hyperbolas of which the eccentricity is 2. This theorem affords a means of trisecting an angle.*

* See Taylor's Ancient and Modern Geometry of Conics, examples 308, 390 with footnote.