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October 9, 2014

Area of Ellipse Problem – Duke Math Meet 2009: Problem 7

Try this problem from Duke Math Meet 2009 Problem 7 based on Area of Ellipse. This problem was asked in the individual round.

Let R_I , R_{II} , R_{III} , R_{IV} be areas of the elliptical region \frac{(x-10)^2}{10} + \frac {(y-31)^2}{31} \le 2009 that lie in the first, second, third, and fourth quadrants, respectively. Find R_I - R_{II} + R_{III} - R_{IV}


Special Note: The answer to this problem is given as 1240. This is the wrong answer. It approximates the region R_I - R_{II} + R_{III} - R_{IV} as a rectangle. However, we provide a solution using symmetry using that assumption and that works fine. Computing area of the ellipse is a tricky business.

First, we draw an approximate picture of the ellipse. The centre is at (10, 31). Here is a figure of it.

ellipse with centre at (10, 31)

ellipse with centre at (10, 31)

To find R_{I} - R_{II} reflect region R_{II} about y-axis. Look at the figure. The shaded region is R_{I} - R_{II} . Its width along the line through centre (y=31) is 20 by symmetry (as the centre is 10 unit away from x-axis so R_I is 20 unit 'thicker' than R_{II} . You may convince yourself about this by solving for x setting y = 31 ).

ellipse 3

reflection of R(II) about the y-axis

Now to find R_{III} - R_{IV} we reflect R_{III} about y-axis again. The strip (shaded in blue) is negative of R_{III} - R_{IV}

-(R3 - R4) shaded in blue

-(R3 - R4) shaded in blue

Note that the blue region 'begins' 31 unit 'below' the minor axis of the ellipse. So if we go 31 unit 'above' the minor axis and take the portion of the red strip, by symmetry it will be equal to the blue strip. We have shaded it in black and red.

-(R3 - R4) in the red strip

-(R3 - R4) in the red strip

So if we want to find R_{I} - R_{II} - { - (R_{III} -R_{IV}) } we remove the black stripe from the red strip and get the final region whose area is R_I - R_{II} + R_{III} - R_{IV} .

R1- R2 +R3 - R4

R1- R2 +R3 - R4

Here is where we apply approximation. Width of the strip is 20 and its height is (31+31) = 62. Hence if we approximate the area as a rectangle, then answer is 1240 (62*20).

But note that the strip is not ACTUALLY a rectangle. So this is only an approximate answer.

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