Find the area between two concentric circles defined by
x2 + y2 -2x + 4y + 1 = 0
x2 + y2 -2x + 4y - 11 = 0
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You can complete the squares in both equations and find the radii of the circles.
Then find the areas of the circles and find the difference.
Do you know how to complete the square?
The equation of a circle with center \((h, k)\) and radius \(r\) is
\((x - h)^2 + (y - k)^2 = r^2\)
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For each equation you are given, you need to complete the square of the x variable and complete the square of the y variable. Then you'll be able to tell what the center is (which we really don't need to know) and what the radius is (which is what we want).
So is the equation (x - 1)2 + (y + 2) 2 = 4 = 22
and for the second one is it (x - 1)2 + (y + 2) 2 = 16 = 42