. Algebraic geometry; a new treatise on analytical conic sections . ow at the centre, the coefficients of x and yare each equal to zero. /. 72a;i-24^1-168 = 0, or 2,x^-y^-1 = Q, (3) and -24a!i+ 582/1+ 106 = 0, or - 12xi + 29yi + 53 = 0 (4) Solving equations (3) and (4) a^ = 2, ^i = - 1-Substituting these values in equation (2) it becomes 36x2 _ 24:xy + 29^^ = y = 0 here, x= ± x = 0 here, y= ± aJ-^ = ± 2 -5 nearly, from a Slide Rule. When x = y, 41a;2 = 180, x = y= ±J^= +21 approx. from a Slide Rule. Now turn the axes through an angle 6, where t-^^=-36^=-T(^^^2 = S- ^^•^•)cos 2
. Algebraic geometry; a new treatise on analytical conic sections . ow at the centre, the coefficients of x and yare each equal to zero. /. 72a;i-24^1-168 = 0, or 2,x^-y^-1 = Q, (3) and -24a!i+ 582/1+ 106 = 0, or - 12xi + 29yi + 53 = 0 (4) Solving equations (3) and (4) a^ = 2, ^i = - 1-Substituting these values in equation (2) it becomes 36x2 _ 24:xy + 29^^ = y = 0 here, x= ± x = 0 here, y= ± aJ-^ = ± 2 -5 nearly, from a Slide Rule. When x = y, 41a;2 = 180, x = y= ±J^= +21 approx. from a Slide Rule. Now turn the axes through an angle 6, where t-^^=-36^=-T(^^^2 = S- ^^•^•)cos 2fl = - Jj, 2 cos^e = 1 - ^\,cos ^ = I, sin 6 = 4. We therefore write - for x, 5 and ^i±^ for 2,.5 The equation becomes 20a;2 +45^2=180, which may be written 9 ^4 The curve is shown in the figure. The particular points determined above are marked with across (X) in the figure. 308 CURVE TRACING. [chap. XIV. 321. Draw the conic W + 8a;y + j/^ + 6a; + 6j/ - 9 = 0. We see that the terms of the second degree have real anddiiferent factors, 1% +1/, and a; + Fig. 180. .. the equation represents a hyperbola. Let («!, 2/i) be the centre, and transfer the origin to that point. AKT. 321.] CURVE TEACING. 309 The equation becomes 1{x + x^f + B{x + X,) {y + y^) + {y +\y + Q{x + x^) + 6(y + y^)-9 = 0 (1) Since the origin is now at the centre, the coefficients of x and yare each equal to zero. .•. Ux^ + 8y^ + & = 0, or 7x^ + 4:y^ + 3 = 0, and 8xi + 2t/i + 6 = 0, or 16a;i +4^1+ 12 = 0. Whence ajj = - 1, and ^j = 1. Substituting these values in equation (1), it becomes 7x + 8xy + y^ = 9 (2) 7a;2 + 8xy + y^ = 0,or {x + tj){7x + y) = 0 represents the asymptotesand we draw these lines. In equation (2), when x = 0, y=±3. „ x = y, x = y=±^=±-75. „ y = 3, x=0, or --V^= - 3-4. „ y= -3, a; = 0 or-2^ = the axes through an angle 6, where cos26 = 1, 2cos26l=l+f, cos 0 = -5=, sm 0 = -=.n/F s/5 We therefore write —p=^ for x, and —j=^ for y. The equation becomes 9x^ - «/
Size: 1341px × 1863px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, bookpublisherlondo, bookyear1916
