anonymous
  • anonymous
Find the area between two concentric circles defined by x2 + y2 -2x + 4y + 1 = 0 x2 + y2 -2x + 4y - 11 = 0
Mathematics
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anonymous
  • anonymous
Find the area between two concentric circles defined by x2 + y2 -2x + 4y + 1 = 0 x2 + y2 -2x + 4y - 11 = 0
Mathematics
chestercat
  • chestercat
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mathstudent55
  • mathstudent55
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.
mathstudent55
  • mathstudent55
Do you know how to complete the square?
mathstudent55
  • mathstudent55
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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mathstudent55
  • mathstudent55
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).
anonymous
  • anonymous
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
anonymous
  • anonymous
I meant \[2^{2}\] and \[4^{2}\]
mathstudent55
  • mathstudent55
\(\color{red}{x^2} + \color{green}{y^2} + \color{red}{(- 2x)} + \color{green}{4y} + 1 = 0 \) \(\color{red}{x^2- 2x} + \color{green}{y^2 +4y} + = -1 \) \(\color{red}{x^2 -2x + 1} + \color{green}{y^2 +4y + 4} + = -1 + \color{red}{1} + \color{green}{4}\) \(\color{red}{(x - 1)^2} + \color{green}{(y + 2)^2} = 2^2\) The first radius is 2.
mathstudent55
  • mathstudent55
\(x^2 + y^2 -2x + 4y - 11 = 0 \) \(x^2 -2x + y^2 + 4y = 11 \) \(x^2 -2x + 1 + y^2 + 4y + 4 = 11+1+4 \) \((x- 1)^2 + (y+ 2)^2 = 4^2 \) The radius of the second one is 4. You are correct again.
mathstudent55
  • mathstudent55
Now find the area of a circle with radius 4 and find the area of a circle with radius 2. Then find the difference of the areas.
mathstudent55
  • mathstudent55
Large circle has radius R and small circle has radius r. \(\large A_{ring} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2)\)
anonymous
  • anonymous
Is it 12 Pi
mathstudent55
  • mathstudent55
\(\large A_{ring} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) =\color{red}{\pi (4^2 - 2^2) = \pi (16 - 4) = 12 \pi} \) You are correct. Great job!
anonymous
  • anonymous
Yay!!

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