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help with this graph

 

Find the area of the region enclosed by the graph of x^2 + y^2 = 2x - 6y + 6 + 12x - 8y + 22.

 Dec 17, 2023
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Step 1: Rewrite the equation

 

First, let's rewrite the equation to a more recognizable form:

 

x^2 + y^2 + 14x - 14y = 28

 

Step 2: Complete the square for both x and y terms

 

To find the area enclosed by the graph, we can try to rewrite the equation as the standard equation of a circle:

 

(x - h)^2 + (y - k)^2 = r^2

 

We can achieve this by completing the square for both the x and y terms:

 

(x^2 + 14x) + (y^2 - 14y) = 28

(x^2 + 14x + 49) + (y^2 - 14y + 49) = 28 + 49 + 49

(x + 7)^2 + (y - 7)^2 = 126

 

Now, we see that the equation represents a circle centered at (-7, 7) with a radius of √126.

 

Step 3: Calculate the area

 

The area enclosed by the circle is simply the area of a circle minus the areas of the four sectors cut off by the coordinate axes:

 

Area = πr^2 - 4 * (1/4) * πr^2

Area = π * √126^2 - 4 * (1/4) * π * √126^2

Area = 126π - 31.5π

Area = 94.5π square units

 Dec 17, 2023

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