Solve 19, 29 and 41.

19-22 Show that the equation represents a sphere, and find its center and radius. 19. x2 + y2 + z2 + 8x - 2z = 8 20. x2 + y2 + z2 = 6x - 4y - 10z 21. 2x2 + 2y2 + 2z2 - 2x + 4y+ 1=0 22. 4x2 + 4y2 + 4z2 = 16x - 6y - 12 23. Midpoint Formula Prove that the midpoint of the line seg- ment from Pi(X1, VI, ZI) to P2(x2, y2, Z2) is X1 + x2 Vi + y2 21+ zz 2 2 2 24. Use the Midpoint Formula in Exercise 23 to find the center of a sphere if one of its diameters has endpoints (5, 4, 3) and (1, 6, -9). Then find an equation of the sphere. 25. Find an equation of the sphere with center (-1, 4, 5) that just touches (at only one point) the (a) xy-plane, (b) yz-plane, and (c) xz-plane. 26. Which coordinate plane is closest to the point (7, 3, 8)? Find an equation of the sphere with center (7, 3, 8) that just touches (at one point) that coordinate plane. 27-42 Describe in words the region of R3 represented by the equation(s) or inequalities. 27. z = -2 28. x = 3 29. y > 1 30. x 2z