(6.) Let x, y, be the co-ordinates of any line, X1, Yı, the co-ordinates of the same line in another system having the same origin, then 1 (x, sin x 7,9 +Y..sin yuy), sin x,y x = N1 If the axės in the original system are rectangular, and in the new system, oblique, the above formulæ become COS X 190 + Y, cos Y198, y=x, sin X1 + y, sin Y1X. If the original axes are oblique, and the new ones rectangular, then 1 (x, sin Q1,4 + y, cos X1,Y), sin x,y sin , If both systems are rectangular, then X=N, COS X 190 - Yı N 1980 y=x, sin x 1,60 + y, cos x 7,8. If it be required to change the origin, as well as the direction of the co-ordinates, then a, b, the co-ordinates of the new origin, must be respectively added to the values of w and y in the above formulæ. If the origin alone be changed, the direction of the coordinates remaining the same, then x= a + x12 y=b+yi (H. A. G. 71-6; L. A. G. . vi; G. G. A. Ch. iii; Biot, 86–96.) (7) Transformation of polar co-ordinates. Let a, b, be the co-ordinates of the new pole, u, the radius vector, and a the angle which the axis to which u is referred makes with the axis of x, then sin {x,y— (@+a)} x=a+u a sin (O + a) y=b+u sin X,Y If the axes are rectangular, these formulæ become x=a + u cos (0 + a), (H. A. G. 77-80; Biot, 100.) THE CIRCLE. (8.) Let a, ß, be the co-ordinates of the centre, r the radius; (x – a)? + (y-2)* + 2(x - a)(y-3).cos ,y=p?. (1) (2) or (x - a)? + yé = pl. If the origin is in the circumference, (1), (2), become respectively, x2 + yo - 2aX – 23y=0); (3) y=2rx - x, do =2g-go. If the origin is at the centre, then 2? + yo =r?. (5) of these forms, (4) and (5) occur most frequently. y? a +1% + The general form when referred to rectangular co-ordinates is yö + x2 + 4y + Bx +C=0. (9.) The polar equation. Let a, b, be the co-ordinates of the pole, u the radius vector, and the angle u, x=0; the polar equation is u° +2 (a cos @ + b sinu + a + b - p=0; + or acting as the pole is situated within or without the circle. (H. A. G. Ch.v; G. G. A. Ch. x; Biot 105-14.) (10.) The intersection of the circle xé + yo=yo and the straight line y=ax + b may be determined from the equation 26 6 - ape? , -0, a +1 which having only two roots, it follows that a straight line cannot cut a circle in more than two points. The equation to a tangent at a given point (x1,9.) in the circumference, is Ny + 91 cos x,y (w — X1), Yit in COS X,Y if the axes are rectangular, y-Y1= (x x1), or Yi DX, +yy;=r? To draw a tangent from a given point (x,y) without the circle: the co-ordinates X1, Yı, of the point of contact may be determined by the solution of the equations, X2X1 + y2y.= gel a + y'=p? (H. A. G. Ch. vi; H. C. S. 37-43; Biot, 115–23.) CONIC SECTIONS. THE PARABOLA. (11.) The parabola referred to its axis. Let S be the focus, P any point in the curve, and PQ perpendicular to EQ the directrix, then PS=PQ. Let PT be a tangent at the point P, and PG the normal. If the origin is at the vertex, and the axis coincides with the axis of x, the equation is yo = 4ma, or PN=4AS. AN. and the parabola yo = 4mx ; may be determined by the solution of the equation 4m 4 mb y + 0: a a which has only two roots; hence a straight line cannot cut a parabola in more than two points. The equation to a tangent at the point (109.) is The subnormal, NG=2 AS=2m. To draw a tangent to the parabola from a given point (a,b) without it: 81,9, the co-ordinates of the point of contact may be determined from the equations 2m y yı? = 4mx1. (H.C. S. 46–62; Hust. Prop. 3—7.) (12.) The parabola referred to the focus. The polar equation, the focus being the pole, is SP 1 + cos e (sin 0) 1 1 2 + (See Fig. p. 160.) SL The tangent at any point, and the perpendicular on it from the focus intersect the axis AY in the same point. SP.SA= SY Let Z be the intersection of the tangent with the directrix, SZ is perpendicular to SP. (H.C. S. 63–71; Hust. Prop. 8.) |