Question

Find the area between two concentric circles defined by
x2 + y2 -2x + 4y + 1 = 0

x2 + y2 -2x + 4y - 11 = 0

Answer 1

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.

Answer 2

Do you know how to complete the square?

Answer 3

The equation of a circle with center \((h, k)\) and radius \(r\) is

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

Answer 4

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).

Answer 5

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

Answer 6

I meant \[2^{2}\]
and \[4^{2}\]

Answer 7

\(\color{red}{x^2} + \color{green}{y^2} + \color{red}{(- 2x)} + \color{green}{4y} + 1 = 0 \)

\(\color{red}{x^2- 2x} + \color{green}{y^2 +4y} + = -1 \)

\(\color{red}{x^2 -2x + 1} + \color{green}{y^2 +4y + 4} + = -1 + \color{red}{1} + \color{green}{4}\)

\(\color{red}{(x - 1)^2} + \color{green}{(y + 2)^2} = 2^2\)

The first radius is 2.

Answer 8

\(x^2 + y^2 -2x + 4y - 11 = 0 \)

\(x^2 -2x + y^2 + 4y = 11 \)

\(x^2 -2x + 1 + y^2 + 4y + 4 = 11+1+4 \)

\((x- 1)^2 + (y+ 2)^2 = 4^2 \)

The radius of the second one is 4.
You are correct again.

Answer 9

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.

Answer 10

Large circle has radius R and small circle has radius r.

\(\large A_{ring} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2)\)

Answer 11

Is it 12 Pi

Answer 12

\(\large A_{ring} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) =\color{red}{\pi (4^2 - 2^2)
= \pi (16 - 4) = 12 \pi} \)

You are correct.
Great job!

Answer 13

Yay!!