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The Ellipse as a Circle with a Moving Center Author(s): F. H. Young Source: The American Mathematical Monthly, Vol. 55, No. 3 (Mar., 1948), pp. 156-158 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2305981 . Accessed: 31/08/2013 00:59 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 205.133.226.104 on Sat, 31 Aug 2013 00:59:45 AM All use subject to JSTOR Terms and Conditions

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Page 1: The Ellipse as a Circle with a Moving Center

The Ellipse as a Circle with a Moving CenterAuthor(s): F. H. YoungSource: The American Mathematical Monthly, Vol. 55, No. 3 (Mar., 1948), pp. 156-158Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2305981 .

Accessed: 31/08/2013 00:59

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access toThe American Mathematical Monthly.

http://www.jstor.org

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Page 2: The Ellipse as a Circle with a Moving Center

156 CLASSROOM NOTES [March,

tion that in all cases where the intercepts exist they are given by

a = p/ cos a and b = p/sin a,

where p is the always positive length of the normal from the origin to the line and a is the positive angle through which the positive x-axis must be rotated to coincide with the positive direction on this normal. Substitution in the inter- cept form yields the normal form in a single step. Lines parallel to one of the axes or passing through the origin require, as usual, special treatment.

Similarly, the normal form of the equation of the plane in space may be de- rived from the intercept form by noting that each intercept (when it exists) is equal to the length of the normal from the origin to the plane divided by the corresponding direction cosine. The analogy may be emphasized by defining, for the two dimensional case, a second angle between the y-axis and the normal to the line.

These derivations are easy for the student to understand and remember. They facilitate unified treatment of plane and solid analytics in an elementary course without requiring explicit discussion of direction cosines in the plane. They lend themselves to efforts toward developing the student's appreciation of mathematical form in general and of the similarities between spaces of dif- ferent dimensions in particular.

THE ELLIPSE AS A CIRCLE WITH A MOVING CENTER

F. H. YOUNG, Oregon State College

Our purpose is to examine the possibility of exhibiting the ellipse as the locus of a point on a rotating circle with a center moving on the focal axis. This condi- tion may be satisfied if we can express the ellipse in the form (x-x')2+y2=b2, for the symmetry of the figure requires that the radius be equal in length to b, half the minor axis. Throughout this discussion we shall assume that a> b.

b b

0 /

Fig. 1

In order to have x' move in such a way as to fit the ellipse x2/a2+y2/b2= 1, we must let x'=kx so that x2(1-k)2/b2+y2/b2 =1 where b2/(1-k)2=a2, whence k = (a-b)/a.

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Page 3: The Ellipse as a Circle with a Moving Center

1948] CLASSROOM NOTES 157

Further, x'=x-b cos 0, but since x'= [(a-b)/a]x, then x=a cos 0. Also, y=b sin 0. Hence x'=kx=(a-b) cos 0. Therefore, (x', 0), the center of the moving circle moves on the focal axis with simple harmonic motion over the range - (a - b) < x' < (a - b) as 0 increases constantly. Conversely, any such moving circle generates an unique ellipse, for the radius is of fixed length, b.

Let us now examine some of the properties revealed by this method of analysis of the ellipse. First, observe that the parametric equations x =a cos 0; y = b sin a can refer to the parameter 0, the angle the moving radius makes with the focal axis.

Consider next the situation arising when the radius is extended in the direc- tion 0+r to give a point (xi, yi) lying the distance b+c from (x, y) on the given ellipse. The point (xi, yi) then has the coordinates yi = - c sin 0 and xi = x'-G Cos 0 = (a - b) cos 0-c cos 0 = (a - b - c) cos 0. Eliminating 0 by squaring and adding, we obtain y2/c2+x2/(a-b-C)2=1, again an ellipse, provided b+cxa. In the particular case in which b+c=a, we find that x=(a-b-c) cos 0=0 and y = (b - a) sin 0, and as 0 goes from 0 to 2wr, we generate that section of the y-axis such that -(a-b) ?y ? (a -b). If b+c>a, and we continue our assumption that a>b, then c>a-b>0: hence |a-b-cl >c and our resulting locus is an ellipse whose major axis lies along the y-axis. It is interesting to note that if c= a, we have our original ellipse with the axes interchanged.

Notice now that the slope of the generating radius is ay/bx. Hence, the radius is normal to the ellipse only at the endpoints of the axes. This result gives the answer to the problem of the type of curve generated by a normal of fixed length N moving about the ellipse. For a fixed N, b -a remains unchanged. Thus, x' moves in the same manner as for the original ellipse, and the radius is now b+N. If (xi, yi) lies on an ellipse, its distance from (x', 0) is b+N, and the line

Fig. 2

joining these points is distinct from the normal. This requires the circle of radius N with center at (xi, y') to be tangent to the ellipse externally in one place and to cut it in another, a contradiction. Hence, the curve generated by the moving normal is definitely not an ellipse. This can similarly be shown if N is measured in from the ellipse.

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Page 4: The Ellipse as a Circle with a Moving Center

158 ELEMENTARY PROBLEMS AND SOLUTIONS [March,

So far we have constrained b to be smaller than a. What results if we have our circle still moving on the x-axis, but b>a? This offers no difficulty, for from the relationship that x' = (a - b) cos 0, we can see that the center of the moving circle now must move from [- (b - a), 0] to [(b - a), 0], as 0 increases positively, in describing the upper half of the ellipse whose equation is x2/a2+y2/b2 = 1, b>a.

ELEMENTARY PROBLEMS AND SOLUTIONS EDITED BY HOWARD EvEs, Oregon State College

Send all communications concerning Elementary Problems and Solutions to Howard Eves, Mathematics Department, Oregon State College, Corvallis, Oregon. This department welcomes problems believed to be new, and demanding no tools beyond those ordinarily fur- nished in the first two years of college mathematics. To facilitate their consideration, solutions should be submitted on separate, signed sheets, within three months after publication of problems.

PROBLEMS FOR SOLUTION

E 806. Proposed by Leo Moser, University of Manitoba

Lewis Carroll once proposed the following problem. "Two travellers spent from 2 o'clock till 9 in walking along a level road,

up a hill, and home again; their pace on the level being x miles per hour, up hill y, and down hill 2y. Find the distance walked."

In the original problem x and y were given integers. Deduce the solution to the original problem without a priori knowledge of what these integers are.

E 807. Proposed by R. V. Andree, University of Wisconsin

An elliptical endgate of a reservoir is to be mounted with the minor axis parallel to the water's surface and in such a manner that it will turn about a horizontal axis in the plane of the gate. Where should this axis be placed so that the gate will not tend to rotate in either direction when the water level is at a given distance above the top of the gate? Generalize to a gate of any shape.

E 808. Proposed by Victor Th1bault, Tennie, Sarthe, France

Show that the number

N = 19000458461599776807277716631

is a perfect cube and that the twenty-eight numbers which are formed by cyclic permutations of its digits are all divisible by the cube root of N.

E 809. Proposed by P. L. Chessin, New York, N. Y.

Show that E,o1 n-(n+l)/n diverges.

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