We can look at examples of quadric surfaces with centers or vertices at points other

Chapter 2, Problem 48

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We can look at examples of quadric surfaces with centers or vertices at points other than the origin by employing a change of coordinates of the form x = x x0, y = y y0, and z = z z0. This coordinate change simply puts the point (x0, y0,z0) of the xyz-coordinate system at the origin of the x yz-coordinate system by a translation of axes. Then, for example, the surface having equation (x 1)2 4 + (y + 2)2 9 + (z 5)2 = 1 can be identified by setting x = x 1, y = y + 2, and z = z 5, so that we obtain x2 4 + y2 9 + z 2 = 1, which is readily seen to be an ellipsoid centered at (1, 2, 5) of the xyz-coordinate system. By completing the square in x, y, or z as necessary, identify and sketch the quadric surfaces in Exercises 4752.z = 4x 2 + (y + 2)2