to either asymptote of the straight lines joining these points respectively to the centre; prove that tan a tan ẞtan y tan 8=1. tangents to the circle described on the straight line joining the foci of the hyperbola as diameter; shew that the locus of x2 y3 1 their poles with respect to the hyperbola is + = b1 a2+b2 a1 26. If two straight lines pass through fixed points, and the bisector of the angle between them is always parallel to a fixed line, prove that the locus of the point of intersection of the lines is a rectangular hyperbola. 27. Shew that pairs of conjugate diameters of an hyperbola are cut in involution by any straight line. 28. The locus of the intersection of two equal circles, which are described on two sides AB, AC of a triangle as chords, is a rectangular hyperbola, whose centre is the middle point of BC, and which passes through A, B, C. CHAPTER VIII. POLAR EQUATION OF A CONIC, THE FOCUS BEING THE POLE. 159. To find the polar equation of a conic, the focus being the pole. Let S be the focus and ZM the directrix of the conic, and let the eccentricity be e. Draw SZ perpendicular to the directrix, and let SZ be taken for initial line. = Let LSL be the latus rectum, then e. SZ SL = 1 suppose. Let the co-ordinates of any point P on the curve be r, 0. Let PM, PN be perpendicular respectively to the directrix and to SZ. If the axis of the conic make an angle a with the initial line the equation of the curve will be For in this case SP makes with SZ an angle - a. 160. If r, directrix, then be the co-ordinates of any point on the 161. To shew that in any conic the semi-latus rectum is a harmonic mean between the segments of any focal chord. If PLP' be the focal chord, and the vectorial angle of S P be 0, that of P' will be +π. Hence, if SP=r, and SP' = r', we have 162. To trace the conic=1+ ecos from its equation. r (1) Let e=1, then the curve is a parabola, and the equation becomes i At the point A, where the curve cuts the axis, As the angle increases, (1 + cos e) decreases, that is decreases, and therefore r increases: and r increases without limit until 07, when r is infinite. As increases beyond π, 1+ cos increases continuously, and therefore decreases continuously until when 0=2π it again becomes equal to 1 2 The curve therefore is as in the figure going to an infinite distance in the direction AS. (2) Let e be less than unity, then the curve is an ellipse. At the point A, 0 = 0, and r = 1+e As increases cose decreases, and therefore - decreases, that is r increases, until 0 = π, r [Since e<1, this value of r is positive.] Z A S P The curve therefore cuts the axis again at some point A' such that SA' = = 1-e As passes from π to 2π, cos increases continuously Since, for any value of 0, cos is symmetrical about the axis. 1+e = cos (2π-0), the curve Therefore when e is less than unity, the equation represents a closed curve, symmetrical about the initial line. (3) Let e be greater than unity, then the curve is an hyperbola. At the point A, 0=0 and r= 1+e |