A first course in projective geometry . nt Taking the conjugate diameters as the axes, this constantis seen to be a? + h^. CARNOTS THEOREM 139 § 9. The Area of the Parallelogram formed by Tan-gents to an Ellipse at the ends of Conjugate Diametersis constant. The area of the projection of a closed figure is equal to thearea of the original figure multiplied by the cosine of theangle between the plane of the figure and the plane of pro-jection (§ 13, Chap. VII.). Apply this to the circle. Since the tangents at 77 and d project into those at P and D,the parallelogram CDTP is the projection of the
A first course in projective geometry . nt Taking the conjugate diameters as the axes, this constantis seen to be a? + h^. CARNOTS THEOREM 139 § 9. The Area of the Parallelogram formed by Tan-gents to an Ellipse at the ends of Conjugate Diametersis constant. The area of the projection of a closed figure is equal to thearea of the original figure multiplied by the cosine of theangle between the plane of the figure and the plane of pro-jection (§ 13, Chap. VII.). Apply this to the circle. Since the tangents at 77 and d project into those at P and D,the parallelogram CDTP is the projection of the square a//jA We have therefore area of parallelogram CDTP = c/;^ x cos 9 = constant for any pair of conjugate diameters CP, CD. If p be the length of the perpendicular from C on the tangentat P, we have therefore area of CDTP =j;.CD = constant, and taking CP and CD as the semi-axes, this constant is seento be ah. § 10. Def. The auxiliary circle is the circle described onthe major (or transverse) axis of a central conic as Fig. 74. 140 PROJECTIVE GEOMETRY Since in the ellipse - = —^j if NP meet the auxiliary • 1 • ^ /-n- »K AC^ -^ circle in Q (Fig. 74), QN^ = AN . NA, PN _BC QN~AC and it follows that Accwdingly the ellipse may he obtained by reducing all (xrdinates of the auxiliary circle in the ratio -. a Ex. 1. The ends of a straight rod move on two fixed straight rodsat right angles. Prove that any point on the moving rod describes anellipse. Ex. 2. A circular disc rolls on the inside of a circular hoop of doubleits radius. Prove that any point on the disc describes an ellipse. No similar property obtains in the case of the hyperbola,as N P does not meet the auxiliary circle. In the case of the parabola, since one vertex is at infinity,the auxiliary circle has infinite radius, and is therefore astraight line—the tangent at the vertex. Historical Note. The discovery of the conic sections isdue to Menaechmus, a pupil of Eudoxus and contemporary ofP
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