. Plane and solid analytic geometry . he locus of the point of intersection, P, of AR and AR ? Choose the center of the circle as the origin and the axis ofX along AA\ Then (1) is the equation of the circle. * If we had taken x as a single auxiliary variable, the coordinates ofR would be {x\ ± \/a2 —x2). t It is possible to represent the motion of i? by a single auxiliary vari-able and at the same time to avoid radicals and preserve symmetry, bychoosing as the auxiliary variable the angle d which the radius drawn toR makes with a fixed direction, the axis of x ; the coordinates of Rare th
. Plane and solid analytic geometry . he locus of the point of intersection, P, of AR and AR ? Choose the center of the circle as the origin and the axis ofX along AA\ Then (1) is the equation of the circle. * If we had taken x as a single auxiliary variable, the coordinates ofR would be {x\ ± \/a2 —x2). t It is possible to represent the motion of i? by a single auxiliary vari-able and at the same time to avoid radicals and preserve symmetry, bychoosing as the auxiliary variable the angle d which the radius drawn toR makes with a fixed direction, the axis of x ; the coordinates of Rare then : x = a cos 6^ y = asin^. We prefer, however, to use as aux-iliary variables the coordinates of R connected by equation (2). 268 ANALYTIC GEOMETRY Take, as the auxiliary motion, the tracing of the circle bythe point It and, as auxiliary variables, the coordinates (x, y) of R. These are connected^ ^ by the equation (2). The ^P:{X Y) coordinates of R are, evi-dently, {x, - y). The equations of ARand AR are 2/-0=-yL(),X -\- a. y 0 y X — a (x— a). Fig. 3 Since P is the point of in-tersection of AR and ARjits coordinates {X, Y) satisfy both these equations: (3)(4) T=-y—{x+d). x-{-a x We have, then, three equations, (2), (3), and (4), involving,besides the constant a, the coordinates (X, Y) of the movingpoint and the auxiliary variables x, y. To obtain an equa-tion in X, Y alone, we must eliminate x, y. We shall do thisby solving two of these equations, preferably (3) and (4),simultaneously for x and y\ and substituting the valuesobtained for them in the third equation, (2). To this end we rewrite equations (3) and (4) as follows: (3a) Yx-{X^a)y = -aY, (4a) Yx + {X- a)y = aY. Hence x = — X 2/ = «^-^ X Substituting these values in (2) and reducing, we obtainas the equation of the locus. A SECOND CHAPTER ON LOCI 269 The locus is thus seen to be a rectangular hyperbola with thegiven diameter of the circle as major axis. It is evident fromthe figure, however, that if
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