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- Consider the ellipsoid x^2+2y^2+3z^2=21.Consider the ellipse E in the xy-plane defined by the equation ax2 + y2 = 1 where a is positive. (1) Find a parametrization r(t) of E (2) Find all the points where r(t) is orthogonal to r'(t).determine a and b as to make the cylinder y^2=4ax orthogonal to the ellipsoid at point (1,2,1)
- find the surfase area of ellipsoid obtained by yevoiving the upper-half of the ellipse x2/a2+y2/b2=1.about x-axis.given that a2-b2=1?Suppose that a cylindrical container of radius r and height L is filled with a liquid with volume V , and rotated along the y-axis with constant angular speed ω. This makes the liquid rotate, and eventually at the same angular speed as the container. The surface of the liquid becomes convex as the centrifugal force on the liquid increases with the distance from the axis of the container. The surface of the liquid is a paraboloid of revolution generated by rotating the parabola y = h + ω2x2/2g around the y-axis, where g is gravitational acceleration and h is shown below. (You can take g=32ft/s2 or 9.8m/s2). Express h as a function of ω. (2) At what angular speed ω will the surface of the liquid touch the bottom? At what speed will it spill over the top? (3) Suppose the radius of the container is 2 ft, the height is 7 ft, and the container and liquid are rotating at the same constant angular speed ω. The surface of the liquid is 5 ft below the top of the tank at the central…Find a parametrization for the cylinder x2 + y2 = 1.
- Find a vector parametric equation for the ellipse that lies on the plane 5x-5y+z=8 and inside the cylinder x2+y2=4. Part C and DThe ellipse x^2/7^2 + y^2/6^2 = 1can be drawn with parametric equations where x(t) is written in the formx(t)=rcos(t) with r = and y(t)Consider a particle traveling clockwise on the elliptical path x2/ 100 + y2/25 = 1. The particle leaves the orbit at the point (−8, 3) and travels in a straight line tangent to the ellipse. At what point will the particle cross the y-axis?
- Match the description of the conic with its standard equation. Hyperbola with vertical transverse axis (x − h)2 a2 + (y − k)2 b2 = 1 (x − h)2 = 4p(y − k) (y − k)2 a2 − (x − h)2 b2 = 1 (y − k)2 = 4p(x − h) (x − h)2 a2 − (y − k)2 b2 = 1 (x − h)2 b2 + (y − k)2 a2 = 12) x2+12xy-4y2 = 30 Just draw by finding the reduced equation (after the required rotation and / or translation) of the cone.