5. Draw the locus of the foci of parabolas passing through two fixed points P and P, and having their axes parallel to a fixed line AB.

[The hyperbola described with P and P, as foci and with length of transverse axis = PM; the side parallel to AB of a right-angled triangle on PP, as hypotenuse.]

6. Given the centre C, vertex A, and a tangent PT meeting CA in T, describe the hyperbola.

[The foot N of the ordinate of the point of contact (P) may be determined from CT: CA :: CA : CN. P is then known. The asymptotes cut off equal distances on PT on each side of P and make equal angles with CA (Prob. 19).]

7. Given the centre C, the axis CT, a tangent PT and its point of contact P, draw the hyperbola. (See last example.)

8. Determine the locus of the intersection of the bisectors of the sides of the triangle formed by the asymptotes and any tangent to a hyperbola.

[A similar and similarly placed hyperbola with axes reduced in ratio 2 : 3.]

9. Given a focus F, tangent PT and point Q on an hyperbola, draw the locus of the second focus.

[From F draw FY perpendicular to PT meeting it in Y: produce FY to ƒ and make Yf=FY. The required locus is the hyperbola with foci ƒ and Q, and transverse axis = FQ.]

10. Given a line QT and two points P and F. From F draw a perpendicular FY to QT meeting it in Y, and produce FY to ƒ, making Yf YF. With P and f as foci and with PF as the distance between the vertices describe an hyperbola. With F and any point on this hyperbola as foci describe an ellipse to pass through P, and shew that it will touch QT:

i.e. Given a focus, tangent and point of a conic, the locus of the second focus is an hyperbola.

11. Given two tangents PT, QT to a rectangular hyperbola and their points of contact P, Q. Shew that if QR be drawn perpendicular to PT, and PR to QP, R will be a point on the

curve.

12. In a given ellipse determine the pair of equal conjugate diameters.

[They coincide with asymptotes of hyperbola having the same axes.]

13. Draw the loci of the points of trisection of a series of circular arcs described on the straight line AB.

[Branches of two hyperbolas having their centres at the internal points of trisection of AB and asymptotes inclined 60° to axis.]

14. Given the asymptotes and a point on a directrix, draw the hyperbola.

15. From a given point P in an hyperbola draw a straight line, such that the segment intercepted between the other intersection with the hyperbola and a given asymptote shall be equal to a given line.

[With P as centre and the length of the given line as radius describe a circle cutting the other asymptote. Either point of intersection joined to P gives the line required.]

16. Given a focus F, and tangent PY to an hyperbola and the length 2a of the transverse axis, shew that the locus of the second focus is a circle.

[From F draw FY perpendicular to PY meeting it in Y; produce FY to ƒ and make Yƒ= FY. ƒ is the centre and 2a the radius of the required circle.]

17. Shew that any point on the circle through the middle points of the sides of a triangle ABC may be taken as the centre of an equilateral hyperbola passing through A, B and C.

18. If four tangents to an equilateral hyperbola be given, shew that either of the limiting points (p. 46) of the system of circles described on the diagonals of the quadrilateral as diameters may be taken as the centre of the hyperbola.

19. Given a focus F, a tangent PT, its point of contact P, and the eccentricity, draw the conic.

[From F draw FT perpendicular to FP and meeting PT in T which will be a point on the directrix. With P as centre and the given eccentricity, describe a

with radius r such that

FP
r

=

circle. Tangents from 7 to this circle will be positions of the directrix. Two solutions are generally possible.]

20. Draw normals to an ellipse, from a given point P.

[The normals pass through the intersections of the ellipse with the rectangular hyperbola passing through P and the centre of the ellipse, and having its asymptotes parallel to the major and minor axes at distances respectively

where a and b are the semi-axes and a, ẞ the co-ordinates of P.]

21. Draw normals to an ellipse from a point on the minor axis.

[They will pass through the intersections of the ellipse with the circle described through the foci and point.]

CHAPTER VII.

RECIPROCAL POLARS AND THE PRINCIPLE OF DUALITY.

In page 31, it has been shewn how to find the pole of a given line and the polar of a given point with regard to a given circle, and the principal properties of poles and polars have been explained.

In pages 140 et seq. an extension has been made to the case of an ellipse, and the properties there noticed are applicable to all conic sections.

The pole of a line with regard to any conic being a point and the polar of a point a line, it follows that any system of points and lines can be transformed into a system of lines and points.

This process is called reciprocation, and it is clear that any theorem relating to the original system will have its analogue in the system formed by reciprocation.

DEF. Being given a fixed conic section (≥) and any curve (S), we can generate another curve (s) as follows; draw any tangent to S, and take its pole with regard to Σ; the locus of this pole will be a curve s, which is called the reciprocal polar of S with regard to Σ. The conic Σ with regard to which the pole is taken is called the auxiliary conic.

A point (of the reciprocal polar curve 8) is said to correspond to a line (of the reciprocated figure S) when we mean that the point is the pole of the line with regard to the auxiliary conic Σ; and since it appears from the definition that every point of s is

the pole with regard to Σ of some tangent to S, this is briefly expressed by saying that every point of s corresponds to some tangent of S.

THEOREM. The point of intersection of two tangents to S will correspond to the line joining the corresponding points of s.

This follows from the property of the conic Σ, that the point of intersection of any two lines is the pole of the line joining the poles of these two lines. (p. 141.)

Now if the two tangents to S be indefinitely near, then the two corresponding points of s will also be indefinitely near, and the line joining them will therefore be a tangent to s; and since any tangent to S intersects the consecutive tangent at its point of contact, the above theorem becomes: If any tangent to S correspond to a point on s, the point of contact of that tangent to S will correspond to the tangent through the point on 8.

Hence we see that the relation between the curves is reciprocal, that is to say, that the curve S might be generated from s (through the auxiliary conic) in precisely the same manner that s was generated from S. Hence the name “reciprocal polars*."

Being given then any theorem of position concerning any curve S (i. e. one not involving the magnitudes of lines or angles), another can be deduced concerning the curve s. For example, if we know that a number of points connected with the figure S lie on a right line, we know also that the corresponding lines connected with the figure 8 meet in a point (the pole of the line with regard to Σ), and vice verså.

From any one such theorem another can be derived by suitably interchanging the words "point" and "line," "inscribed" and "circumscribed," "locus" and "envelope," &c., understanding by the term envelope "the curve to which a series of lines drawn according to any given rule are tangents."

The evolute of a curve, e. g. is the envelope of normals to the

curve.

*Salmon's Conic Sections, chap. xv.