Question 1 Consider a sphere S with center - 1 , 2 , 1 and...

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Northwestern University NetID: ID # : Math 230 Final Exam Solutions!!!! Spring Quarter 2013 Tuesday, June 11, 2013 Complete the front page!!!! [ 2 points! ] Instructions: Show ALL of your work on the pages provided! Explain your reasoning using complete sentences and correct grammar, spelling, and punctuation. All numerical answers MUST be exact; e.g., you should write π instead of 3 . 14 ... , 2 instead of 1 . 414 ... , and 1 3 instead of 0 . 3333 ... Make sure that your final answer is clearly indicated. This test has fifteen pages. Check to make sure you have pages 1 - 15. Calculators, course notes, textbooks, etc. are NOT allowed. Do not write your name on this exam! Write your NetID and your Student ID Number in the spaces provided at the top of this page. Mark your section!! X Sec. # Time Instructor 31 9:00 Belt 41 10:00 Belt 51 11:00 Lizzie 61 12:00 Goldstein 71 1:00 Garton 81 2:00 TiM Question Possible Points Score Front Page 2 1 8 2 6 3 8 4 6 5 11 6 5 7 8 8 8 9 10 10 8 11 10 12 10 Total 100
Math 230 Final Exam Solutions! Spring Quarter 2013 Page 2 of 15 Question 1. Consider a sphere S with center ( - 1 , 2 , 1) and passing through the point (6 , - 2 , 3). (a) (3 points) Find an equation of the sphere.
(b) (2 points) Find an equation of the line through (6 , - 2 , 3) and the center of the sphere.
(c) (3 points) Two points P and Q on a sphere are called antipodal points if the line segment between P and Q passes through the center of the sphere 1 . Find the point Q on the sphere such that (6 , - 2 , 3) and Q are antipodal points.
Math 230 Final Exam Solutions! Spring Quarter 2013 Page 3 of 15

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