Science:Math Exam Resources/Courses/MATH200/December 2011/Question 04
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 • Q5 (a) • Q5 (b) • Q6 • Q7 • Q8 (a) • Q8 (b) i • Q8 (b) ii • Q8 (b) iii •
Question 04 

Find the radius of the largest sphere centred at the origin that can be inscribed inside (that is, enclosed inside) the ellipsoid 
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 

If the sphere with radius in question is enclosed by the ellipse, and the radius is maximal, then the surface of the ellipse and the sphere just touch in only one point. What must hold for the tangent planes to the surface and the sphere with radius at the points where the surface and the sphere touch? 
Hint 2 

In the points where the ellipsiod and the sphere touch in only one point, the surfaces are parallel. Hence, the normal vector to the surfaces are parallel in these points, too.

Hint 3 

The function are
Hence, and

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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. If the sphere with radius in question is enclosed by the ellipse, and the radius is maximal, then the surface of the ellipse and the sphere just touch in one point. The tangent planes to the surface and the sphere with radius are parallel at the points where the surface and the sphere touch. Hence the normal vectors to the tangent planes are also parallel. Regarding the surface equation and the sphere equation as level sets of functions , then the normal vectors to these tangent planes are given by the gradients of the functions and . The sphere equation with the origin as center and radius is . Then and . We are looking for such that the gradients are parallel,
The second line yields that or .
These four points are candidates for where the ellipse and the sphere touch. Since we are searching for the radius such that the sqhere is enclosed in the ellipse, we need the point with the shortest distance to the origin. Since is the smallest of these numbers, this is the radius of the sqhere. 