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Determine the value of A1 - A2 + A3 - A4

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anemone

MHB POTW Director
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Feb 14, 2012
3,755
Let \(\displaystyle A_1\), \(\displaystyle A_2\), \(\displaystyle A_3\) and \(\displaystyle A_4\) represent the areas within the ellipse \(\displaystyle \frac{(x-19)^2}{19}+\frac{(y-98)^2}{98}=1998\) that are in the first, second, third and fourth quadrants respectively. Determine the value of \(\displaystyle A_1-A_2+A_3-A_4\)
 
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anemone

MHB POTW Director
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Feb 14, 2012
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Re: Determine the value of A1-A2+A3-A4

I must mention here that the idea of this problem came from Professor Gregory Galperin of Eastern Illinois State University, U.S.A. and I feel so grateful and thankful to Professor Gregory Galperin for approving me to show his solution to this problem at this site.

Solution:

We know that the center of the ellipse is the point (19, 98). The two vertical lines \(\displaystyle x=0 \) and \(\displaystyle x=2\cdot19=38\) and the two horizontal lines \(\displaystyle y=0\) and \(\displaystyle y=2\cdot98=196\) divide the ellipse into nine regions, as shown below.

ellipse.JPG

Then, in terms of the areas marked by S, T, U and V, we find that \(\displaystyle A_1=U+V+S+T\), \(\displaystyle A_2=S+T\), \(\displaystyle A_3=S\), and \(\displaystyle A_4=V+S\), and therefore

\(\displaystyle A_1-A_2+A_3-A_4=(U+V+T+S)-(S+T)+S-(V+S)=U.\)

Hence the answer is \(\displaystyle 38\cdot196=7448\).


Note:
This solution was published in the book by G. Berzsenyi "International Mathematics Talent Search, Part 2 ", AMT Publishing(2011). The problem was given at the Round 29, and it's titled as Problem 5/29; the pages are: the formulation @ p.11, and the solution @ p.84.