## anonymous one year ago Find the area between two concentric circles defined by x2 + y2 -2x + 4y + 1 = 0 x2 + y2 -2x + 4y - 11 = 0

1. 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.

2. mathstudent55

Do you know how to complete the square?

3. mathstudent55

The equation of a circle with center $$(h, k)$$ and radius $$r$$ is $$(x - h)^2 + (y - k)^2 = r^2$$

4. 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).

5. 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

6. anonymous

I meant $2^{2}$ and $4^{2}$

7. 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.

8. 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.

9. 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.

10. mathstudent55

Large circle has radius R and small circle has radius r. $$\large A_{ring} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2)$$

11. anonymous

Is it 12 Pi

12. 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!

13. anonymous

Yay!!