Question

In Exercises 39-48, find a parametrization of the curve. 39. The vertical line passing through the point (3,2,0) 40. The line passing through (1,0,4) and (4.1.2) 41. The line through the origin whose projection on the xy-plane is a line of slope 3 and whose projection on the yz-plane is a line of slope 5 (ie, Az/Ay = 5) 42. The circle of radius 1 with center (2, -1, 4) in a plane parallel to the xy-plane 43. The circle of radius 2 with center (1,2,5) in a plane parallel to the yz-plane 44. The ellipse = 1 in the xy-plane, translated to have cen- ter (9.-4,0) 45. The intersection of the plane y = with the sphere x² + y2 + 2 = 1 46. The intersection of the surfaces z = x2 - y2 and z=r?+xy - 1 in the xz-plane, translated to have cen- 47. The ellipse ter (3, 1,5) (Figure 14(A)] 1, translated to have center (3,1,5) 48. The ellipse [Figure 14(B)]
#48

#46 and #48

Answer 1

(46) = x² - y² and z= x² +ay-1 egnate both equations - x² - y2 = y²+ay-1 - 0 = y²tay- Plet y = t), > a = + -t) and ay = 1-t² Z: x2 + xy- > z= (t -t) 4 (1-42) - 1 +442(- 1) - 2 (txt) + 1-4²_1 РNo

There for parametersation's [814)= < (7-7), t, (+2 -2) > S=1 center (3, 1,5) [x-3 Ixfixed and translated to have conter (3,11). new ellipse equation is > (3-1=431-605)) 9:14 a 191215 -108) J = 1+ 259 2²_254102 92-25 +0Z 13² 17 2 10z - 2²_16 Put , 2 = 1), a = 1 + 3 Jiot - 1²16 ) - Parameterisation is 84).<3, 023 - Cro).ts TUN