(6.) Let x, y, be the co-ordinates of any line, 1, y1, the co-ordinates of the same line in another system having the same origin, then If the axes in the original system .are rectangular, and in the new system, oblique, the above formulæ become If the original axes are oblique, and the new ones rectangular, then 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 x and y in the above formulæ. If the origin alone be changed, the direction of the coordinates remaining the same, then x = a + x 19 (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 x = a + u y=b+u sin {x,y — (0 + a)} sin (0+ a) sin x,y If the axes are rectangular, these formulæ become x=a+u cos (0 + a), y=b+u sin (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; then the equation to the circle is (x a)2 + (y − ẞ)2 + 2 (x − a) (y − ẞ). cos x, y = r2. The equation between rectangular co-ordinates is (x − a)2 + (y — ß)2 = r2. (1) If the axis of x, or y, passes through the centre, the equation becomes respectively, (1), If the origin is in the circumference, (1), (2), become of these forms, (4) and (5) occur most frequently. is The general form when referred to rectangular co-ordinates y2 + x2 + Ay+B+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 u2±2 (a cos 0 + b sin 0) u + a2 + b2 — r2 = 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 x2 + y2= r2 and the straight line yax+b may be determined from the equation 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 (x,y,) in the circumference, is To draw a tangent from a given point (x,y) without the circle: the co-ordinates 1, y1, of the point of contact may be determined by the solution of the equations, (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 PSPQ. 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 a, the equation is may = The intersection of the straight line y = ax + b, and the parabola y2=4mx; be determined by the solution of the equation 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 (x,y,) is The subnormal, NG=24S=2m. To draw a tangent to the parabola from a given point (a,b) without it: x,y, the co-ordinates of the point of contact may be determined from the equations The polar equation, the focus being the pole, is The tangent at any point, and the perpendicular on it from the focus intersect the axis AY in the same point. SP.SA-SY2. Let Z be the intersection of the tangent with the directrix, SZ is perpendicular to SP. (H. C. S. 63–71; Hust. Prop. 8.) X |