Plane and solid analytic geometry; an elementary textbook . mbine the two previous formulas into the one set, , . , , M 05 = OCo 4- oc cos 0-2/ Sill 0, r221 y = 2/o + x sin 0 + y cos 0. *--*J PROBLEMS 1. Transform the equation 3 x + 7y = 8 to a new set ofaxes parallel to the old set, and having the point (4, — 2)as origin. 2. Show that the equation x2 + y2 = a2, referred to rectan-gular axes, will be unchanged by revolving the axes throughany angle, keeping the origin fixed. 3. Transform the equation x2 — y2 = 10, referred to rectangular axes, to axes bisecting the angle between the old axes.
Plane and solid analytic geometry; an elementary textbook . mbine the two previous formulas into the one set, , . , , M 05 = OCo 4- oc cos 0-2/ Sill 0, r221 y = 2/o + x sin 0 + y cos 0. *--*J PROBLEMS 1. Transform the equation 3 x + 7y = 8 to a new set ofaxes parallel to the old set, and having the point (4, — 2)as origin. 2. Show that the equation x2 + y2 = a2, referred to rectan-gular axes, will be unchanged by revolving the axes throughany angle, keeping the origin fixed. 3. Transform the equation x2 — y2 = 10, referred to rectangular axes, to axes bisecting the angle between the old axes. 72 ANALYTIC GEOMETRY [Cn. VI, §§ 49, 50 4. Through what angle must the coordinate axes be turned,if in its new position the X-axis goes through the point (5, 7) ? 5. Given the equation xr + y2 -4- 8 x — ky = 0. To- whatpoint must the origin be changed to cause the terms in x and yto disappear ? 6. Given the equation 2y2 + 2 xy + r + 4 = 0, referred torectangular axes. Through what angle must the axes beturned to cause the term in xy to disappear ?. Fig. 44. % Let the student show that 49. Transformation fromany Cartesian system to anyother Cartesian system, hav-ing the same origin. — InFig. 44, OX and OY arethe original axes, and co isthe angle between them ;OX and OY are the newaxes; OX and OY makeangles 6 and cj> with OX. sin w r/sinO-4>)3sinw sin w ySin*. [23] sin « What do these formulas becotne when co — 90° ? Whenco = 90° and $ = 6? 50. Degree of an equation not changed by transformationof coordinates. — The degree of an equation cannot bechanged by transformation from one system of Cartesiancoordinates to any other. For we have seen that in eachcase we replace x and y by expressions of the first degree Ch. VI, §51] TRANSFORMATION OF COORDINATES 73 in xf and y\ and that therefore the degree of the equationcannot be raised. Neither can it be lowered, for it wouldthen be necessary to raise the degree in transformingback to the original axes, since we
Size: 1735px × 1440px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1900, bookpublishernewyorkcscribnerss
