Generate an appealing image showing a geometric figure. The figure should showcase a square ABCD with a side length illustrated to emphasize that it is of length 60 units. This square should be completely circumscribed by an ellipse E. Distinctly mark a point P on the ellipse and show that the distances from P to points C and D on the square are of equal length, represented as 50 units. Notably, do not include the point (30, 30) in the illustration or any provided text.

Square ABCD has side length 60. An ellipse E is circumscribed about the square and there is a point P on the ellipse such that PC = PD =50. What is the area of E?

I got to the part with the point (30, 30). Now what next?

If you draw the figure, you will see that the ellipse has semi-major axis of 30+40=70

So, the equation of the ellipse is

x^2/70^2 + y^2/b^2 = 1

Since (30,30) is on the ellipse,

30^2/70^2 + 30^2/b^2 = 1
b = 21√(5/2)

The area of the ellipse is thus

A = πab = 735π√10

To find the area of the ellipse E, we first need to determine the semi-major and semi-minor axes.

Given that square ABCD has a side length of 60, the diagonal of the square is √(60^2 + 60^2) = √(2 * 60^2) = √(2) * 60.

Since the diagonals of a square are also the diameters of the circumscribed ellipse, the length of the semi-major axis (a) of the ellipse is equal to half of the length of the diagonal of the square, which is (√(2) * 60)/2 = √2 * 30.

Next, we need to find the length of the semi-minor axis (b) of the ellipse. Since the square is inscribed in the ellipse, the length of the semi-minor axis is equal to half of the side length of the square, which is 60/2 = 30.

Now that we have the values of a and b, we can use the formula to calculate the area of the ellipse: Area = π * a * b.

Substituting the values, we have Area = π * √2 * 30 * 30 = 900π√2.

Therefore, the area of the ellipse E is approximately 900π√2.

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735π√10