# Full text of "A Method for Constructing an Hyperbola, Given the Asymptotes and a Focus"

## See other formats

STOP Early Journal Content on JSTOR, Free to Anyone in the World This article is one of nearly 500,000 scholarly works digitized and made freely available to everyone in the world by JSTOR. Known as the Early Journal Content, this set of works include research articles, news, letters, and other writings published in more than 200 of the oldest leading academic journals. The works date from the mid-seventeenth to the early twentieth centuries. We encourage people to read and share the Early Journal Content openly and to tell others that this resource exists. People may post this content online or redistribute in any way for non-commercial purposes. Read more about Early Journal Content at http://about.jstor.org/participate-jstor/individuals/early- journal-content . JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. 285 A simple modification leads to a construction for a tangent to a parabola from any external point G. We have only to replace the directing circle by the directrix of the parabola. The Hyperbola. Proposition VI. If at two fixed points P and P, three lines A, B, and G, be pivoted, A at one point revolving in one direction at any velocity; B and G at the other pivot revolving in an opposite direction, G at such a rate that it constantly intersects A in the circumference of a directing circle described with P as a center, B at such a rate that the angle BG is constantly equal to the angle GA, then the locus of the intersection M of A and B is a hyperbola. Let the angle AG be denoted by (j> and BG by 8. Since ^=#, the segment JOf =segment PM in any j position. Therefore PM - PM = PM— NM=PW= | constant. Therefore the locus of M is a hyperbola. Proposition VII. If a circle with center G be I described in the plane of a hyperbola passing through one focus P and intersecting the directing circle at H, and the other focal radius P' Mbe drawn through this point H to meet the curve, at M, the line GM is tangent to the hyperbola. Draw MP and HP. The triangle BMP is isos- celes and GM is perpendicular to the base PU at its mid-point A. Therefore it passes through the vertex M and is tangent to the hyperbola. A METHOD FOR CONSTRUCTING AN HYPERBOLA, GIVEN THE ASYMPTOTES AND A FOCUS. By ARCHIBALD HENDERSON, Ph. D„ Associate Professor of Mathematics, University of North Carolina, Chapel Hill, N. C. Consider any circle, whose center is the point (0, y g ) and whose radius is the distance from this point to the focus £i/(a 2 +& 3 ), 0] of an hyperbola. The equation of this circle is or^+y 8 -2y y-(a*+& 8 )=0....(l). Now we may represent any point on an asymptote to the hyperbola 286 a 2 b* i—W by introducing the parameter t. Thus (a;,, y i )=(at 1 , bt^) represents any point on the asymptote 2,-A^O....(3), and (x. x , y t )= (— at it bt t ) represents any point on the asymptote y+-|-*=0....(4). If the circle (1) cuts the asymptotes (3) and (4) in the specified points (^n tti)> ( x i> Hz)) respectively, we have (a*+b*Xt?-l)=2by t 1 ....(5), (a 2 +&*)(* 8 2 -l)=2ty * 8 ....(6). By division we obtain which may be written (< 1 -< i )(«,< i +l)=0....(7). The solution <,-< t =0....(8) shows that, for one position of (a 2 , # 2 ), the line joining (a;, , y, ) and (x 2 , y 2 ) is parallel to the z-axis. Discarding this case, let us consider the solution * 1 «„+1=0....(9). Since (x t , «/ 2 )=(-^-, —. — ), the equation of the line joining («,,y,) and y—bt 1 x—at-i or But this line touches the hyperbola (2), since 287 4J>H* b* mr- &«.... (ii). TTie lines joining the pairs of points {right hand, say) in which a system of co- axial circles, passing through the foci of an hyperbola, cuts the asymptotes, envelope that hyperbola. Since, moreover, the middle point of the line joining (x x , «/,), (a; 2 , t/ 8 ) lies on the hyperbola, we have the theorem : The middle points of the line? join- ing the pairs of points in which a system of I co-axial circles, passing through the foci of j an hyperbola, cuts the asymptotes, describe that hyperbola. These two theorems give two methods for constructing an hyperbola, the one by lines, the other by points, when the asymptotes and a focus are known.* Other constructions might readily have been given, but those given above seem the most instructive. The University of Chicago, November, 1902. * Compare the November number of the Monthly for a note by the writer on the converse of this problem. DEPARTMENTS. SOLUTIONS OF PROBLEMS. ARITHMETIC. 163. Proposed by CHRISTIAN HORNUNGf, A.M.. Professor of Mathematics, Heidelberg University, Tiffin. 0. Three Dutchmen and their wives went to market to buy hogs. The names of the men were Hans, Klaus, and Hendricks, and of the women, Gertrude, Anna, and Katrine; but it was not known which was the wife of each man. They each bought as many hogs as each man or woman paid shillings for each hog, and each man spent three guineas more than his wife. Hendricks bought 23 hogs more than Gertrude, and Klaus bought 11 more than Katrine. What was the name of each man's wife? Solution by J. SCHEFFER. A. M„ Hagerstown, Md.. and M. E. GEABER, Heidelberg University, Tiffin, 0. Let x represent the number of one of the women's hogs, and?/ the number of her husband's; then by the conditions of the problem j/ a =a; 2 +63. Conse- quently a; 2 +63 must be an integer, since |/(a: 2 +63) represents the number of hogs. The equation y 2 — <c s =63 or (y-\- x)(y— x)—63 admits of three solutions, viz., 63x1, 21x3, and 9x7.