From equation to area

Geometry Level pending

An ellipse is specified by its equation:

100 x 2 + 36 y 2 − 1000 x + 576 y + 1204 = 0 100 x^2 + 36 y^2 - 1000 x + 576 y + 1204 = 0

Its area is n π n \pi for a positive integer n n . Find n n


The answer is 60.

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Equation of that ellipse 100 x 2 + 36 y 2 − 1000 x + 576 y + 1204 ⟹ 25 ( x − 5 ) 2 + 9 ( y + 8 ) 2 = 900 100x^2+36y^2 -1000x+576y+1204 \implies 25(x-5)^2+9(y+8)^2= 900 ⟹ ( x − 5 ) 2 36 + ( y + 8 ) 2 100 = 1 \implies \dfrac{(x-5)^2}{36} + \dfrac{(y+8)^2}{100} = 1 Area of a general ellipse x 2 a 2 + y 2 b 2 = 1 \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} = 1 is π a b πab

Here a = 6 a=6 , b = 10 b=10 , so area = 60 π =60π

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