Find the area of the region that lies outside the circle x^2 + y^2 = 4 but inside the circle x^2 + y^2 -6 x -16 = 0.
x^2 + y^2 -6 x -16 = 0
x^2 - 6x + 9 + y^2 = 16+9
(x-3)^2 + y^2 = 25
centre at (3,0), radius 5
x^2 + y^2 = 4
centre at (0,0), radius 2
a quick sketch will show that the smaller circle lies entirely inside the larger
and they have a common point at (-2,0)
So all we need is the area of the larger - area of the smaller
= 25π - 4π square units
= 21π square units
To find the area of the region that lies outside the circle x^2 + y^2 = 4 but inside the circle x^2 + y^2 - 6x - 16 = 0, we need to subtract the area of the smaller circle from the area of the larger circle.
Step 1: Identify the radii of both circles:
The equation x^2 + y^2 = 4 has a radius of sqrt(4) = 2.
The equation x^2 + y^2 - 6x - 16 = 0 can be rewritten as (x - 3)^2 + y^2 = 25. This equation has a radius of sqrt(25) = 5.
Step 2: Calculate the area of each circle:
The area of a circle is given by the formula A = πr^2.
The larger circle: A1 = π(5)^2 = 25π.
The smaller circle: A2 = π(2)^2 = 4π.
Step 3: Subtract the smaller circle area from the larger circle area:
Area of the region between the two circles = A1 - A2 = 25π - 4π = 21π.
Therefore, the area of the region that lies outside the circle x^2 + y^2 = 4 but inside the circle x^2 + y^2 - 6x - 16 = 0 is 21π.
To find the area of the region that lies outside the circle x^2 + y^2 = 4 but inside the circle x^2 + y^2 - 6x - 16 = 0, we can use the concept of the equation of a circle.
First, let's find the equation of the circle x^2 + y^2 = 4. This equation represents a circle centered at the origin (0,0) with a radius of 2.
Next, let's find the equation of the circle x^2 + y^2 - 6x - 16 = 0. We can rewrite this equation as (x - 3)^2 + y^2 = 25. This equation represents a circle centered at (3,0) with a radius of 5.
The area we want to find is the region that lies outside the smaller circle (x^2 + y^2 = 4) but inside the larger circle ((x - 3)^2 + y^2 = 25).
To find the area of this region, we can subtract the area of the smaller circle from the area of the larger circle.
The area of a circle can be found using the formula A = πr^2, where A is the area and r is the radius.
For the smaller circle, A1 = π(2)^2 = 4π.
For the larger circle, A2 = π(5)^2 = 25π.
To find the area of the region that lies outside the smaller circle but inside the larger circle, we subtract A1 from A2:
A = A2 - A1 = 25π - 4π = 21π.
Therefore, the area of the region that lies outside the circle x^2 + y^2 = 4 but inside the circle x^2 + y^2 - 6x - 16 = 0 is 21π square units.