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Find the area of the region enclosed by the graph of $x^2 + y^2 = 2x - 6y + 6 + 14x - 16y + 80$.

 Jan 12, 2024
 #1
avatar+288 
+1

\(x^2 + y^2 = 2x - 6y + 6 + 14x - 16y + 80\)

 

Graph:

 

 

We can find the circle radius and we see that it is:

 

 \(\sqrt{271}\)

 

Area: \(\pi r^2\)

 

Area = \(271\pi\), or \(851.371609123\)

 

Answer: 271π units squared or 851.37 units squared

 Jan 12, 2024
 #2
avatar+128448 
+1

Simplify  as

 

x^2 - 16x + y^2 + 22y  =  80   complete the square on x, y

 

x^2 -16x + 64  + y^2 + 22y + 121  =   80 + 64 + 121

 

(x - 8)^2   +   (y + 11)^2   =  265

 

The center is (8,  -11)     and the area  is   265 pi ≈  832.5

 

cool cool cool

 Jan 12, 2024

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