Plane and solid analytic geometry; an elementary textbook . Fig. 67. 122 ANALYTIC GEOMETRY [Ch. IX, § 72 Points on the ellipse and auxiliary circle which havethe same abscissa are called corresponding points. In. Fig. 68. Fig. 68, Px and R are corresponding points. The angleMXCR is caHed the eccentric angle of the point Pv There is a simple relation between the ordinates of thecorresponding points P1 and R which may be found inthe following manner. Let the coordinates of Pl be(xv y^, and of R (xv ?/2). Substituting these values for x and y in the equationsof the ellipse and circle respectively
Plane and solid analytic geometry; an elementary textbook . Fig. 67. 122 ANALYTIC GEOMETRY [Ch. IX, § 72 Points on the ellipse and auxiliary circle which havethe same abscissa are called corresponding points. In. Fig. 68. Fig. 68, Px and R are corresponding points. The angleMXCR is caHed the eccentric angle of the point Pv There is a simple relation between the ordinates of thecorresponding points P1 and R which may be found inthe following manner. Let the coordinates of Pl be(xv y^, and of R (xv ?/2). Substituting these values for x and y in the equationsof the ellipse and circle respectively, we have b2x^ + a2y12 = a2b2,and x? -f- y2 = a2. Multiplying the second equation by b2 and subtracting,we have or a2y2 = b2y2, v±=± _. y2 a Or, the ordinate of any point of the ellipse is to the ordinateof the corresponding point of the circle as b is to a. Ch. IX, § 73] CONIC SECTIONS 123 Let the student show that a similar relation holdsbetween the abscissas of points on the ellipse and minorauxiliary circle which have the same ordinate. These arealso called corresponding points. Show that CR passesthrough the corresponding point in the minor auxiliarycircle. PROBLEMS 1. Find the focal radii of the
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