Science:Math Exam Resources/Courses/MATH253/December 2012/Question 01 (b)
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Question 01 (b) 

Let P be the plane tangent to the ellipsoid at the point (1,1,2). Find the point where P intersects the yaxis. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

The plane equation is 
Hint 2 

Once you calculated the equation for the plane, set x = z = 0 to find the intersection with the yaxis. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let . So, Plugging in the point (1,1,2), the gradient is . Using this in the plane equation: we get Since we want to find the intersection of this plane with the yaxis, we know that x = z = 0. The equation then becomes Solve for y: Collect like terms: 