What type of three-dimensional object will form from rotating this two-dimensional object around its long edge

The plane x+y+2z=4 intersects the paraboloid z=x^2+y^2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin

The left panel of Figure 3 shows an elliptic hyperboloid of two sheets given by 4x² - y² - 2² = 1. The middle panel shows the plane 2x space curves. With the plane removed for clarity, the right panel just shows the space curves and the hyperboloid.The magenta and navy space curves will be referred to as C₁ and C₂ respectively. y + z = k intersecting the two sheets, resulting in two -10 -10 -10 10 10 10 10 10 Zo -10 -10 0 10 Figure 3 The parametrisation used to plot these space curves is 2t2-4t+5 -10 10 Zo 10 -10 10 0 10 x(t) y(t) 4(t - 2) = t z(t) = 3- 4t 2(t-2) a) Find the domain of t for C₁ and C2 (1) དྱེསྱེºེ (2) (3) b) Using equations (1), (2) and (3) as a guide, determine the general space curve parametrisation in terms of k for the case presented above. This doesn't need to involve the 3 equations subbed directly into the hyperboloid/plane equations, just i.e sub the plane equation into the hyperboloid equation, leaving a relationship involving x,y and k; or y,z and k. Also,…