A first course in projective geometry . ote 2. If / is the length of the latus rectum, we haveproved that in the parabola PN- = L AN. In the central conic 1 = — (§ 3). ^ ^ PN2 62 I But , = + -„ = + —. -^2 -2a PN2 NA = + LAN - 2aThis is in both cases a positive ratio, less than unity in theellipse, and greater than unity in the hyperbola. We havethen PN2 = Z. AN, according as the conic is an ellipse, parabola,or hyperbola. § 5. The fundamental Focus-Directrix Property. Prop. If any point P be taken on a conic and PK beperpendicular to that directrix which corresponds to a focus S,SP/PK is
A first course in projective geometry . ote 2. If / is the length of the latus rectum, we haveproved that in the parabola PN- = L AN. In the central conic 1 = — (§ 3). ^ ^ PN2 62 I But , = + -„ = + —. -^2 -2a PN2 NA = + LAN - 2aThis is in both cases a positive ratio, less than unity in theellipse, and greater than unity in the hyperbola. We havethen PN2 = Z. AN, according as the conic is an ellipse, parabola,or hyperbola. § 5. The fundamental Focus-Directrix Property. Prop. If any point P be taken on a conic and PK beperpendicular to that directrix which corresponds to a focus S,SP/PK is a constant ratio (called the Eccentricity). Let P be any point on the curve and let PA, PA meet thedirectrix in E, E (Fig. 77). THE FOCI 151 Then SE, SE are conjugate lines at right angles. It has been proved that if ES meets AP in V, EPVA is aharmonic range. The pencil obtained by connecting thisrange to S has two conjugate rays at right angles ; there-fore these rays bisect the angles between the other pair ofconjugate Fig. 77. .*. angle PSV = angle ASV = angle PLS if SV meets KP in L. .. SP = PL. Hence SP/PK = LP/PK = SA/AX. Similarly if SE meets PK in L, PL = PS, and either of theabove ratios is equal to SA/AX. The student should draw the figure for the case of thehyperbola. § 6. The Eccentricity in terms of the Semi-Axes. Denote the eccentricity by e. SA + SA SA - SA Then But SA^ = AX = SAAX AX + AX AX - AX rsA± lSA + ±SA = 2CA, AX±AX = 2CXSA = 2CS, AX + AX = 2CA )• 152 PROJECTIVE GEOMETRY the upper signs in each case referring to the ellipse and thelower to the hyperbola, and the absolute magnitudes of thelengths concerned alone being taken into account. CS CA . , .,• • 6 = ;^ = ;^T7 in l>otn CX dq2 RC^ .-. by § 2, c = 1 - ^ (ellipse) or 1 + ^ (hyperbola). e is therefore less than 1 in the former case and greaterthan 1 in the latter. For the parabola, since SA = AX, 6 = 1. From Fig. 76 we have then SP=PK, where P is any pointon the c
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