The area bounded by circles `x^2 +y^2=r^2`, r = 1, 2 and rays given by `2x^2-3xy-2y^2=0`,is

Options

A. `pi //4` sq. units
B. `pi //2` sq. units
C. `3pi//4` sq. units
D. `pi ` sq. units

Answer

To find the area bounded by the circles \(x^2 + y^2 = r^2\) for \(r = 1\) and \(r = 2\), and the rays given by \(2x^2 - 3xy - 2y^2 = 0\), we will follow these steps:

### Step 1: Analyze the given equation of the rays
The equation \(2x^2 - 3xy - 2y^2 = 0\) can be factored to find the slopes of the lines (rays).

### Step 2: Factor the equation
We can rewrite the equation as:
\[ 2x^2 - 3xy - 2y^2 = 0 \]
Factoring this, we can rearrange it:
\[ (2x + y)(x - 2y) = 0 \]
This gives us two lines:
1. \(y = -2x\) (from \(2x + y = 0\))
2. \(y = \frac{x}{2}\) (from \(x - 2y = 0\))

### Step 3: Identify the circles
The circles are given by:
1. \(x^2 + y^2 = 1\) (for \(r = 1\))
2. \(x^2 + y^2 = 4\) (for \(r = 2\))

### Step 4: Sketch the circles and lines
- The first circle (radius 1) is centered at the origin and has a radius of 1.
- The second circle (radius 2) is also centered at the origin and has a radius of 2.
- The lines \(y = -2x\) and \(y = \frac{x}{2}\) will intersect the circles.

### Step 5: Find the area between the circles and the lines
The area bounded by the circles and the lines can be calculated using the formula for the area of a circle minus the area of the smaller circle.

The area \(A\) of a circle is given by:
\[ A = \pi r^2 \]
For the larger circle (radius 2):
\[ A_2 = \pi (2^2) = 4\pi \]
For the smaller circle (radius 1):
\[ A_1 = \pi (1^2) = \pi \]

### Step 6: Calculate the area between the two circles
The area between the two circles is:
\[ A_{\text{bounded}} = A_2 - A_1 = 4\pi - \pi = 3\pi \]

### Step 7: Adjust for the area outside the rays
Since the rays divide the area into sections, we need to account for the area that is outside the rays. The area between the rays and the circles can be calculated as:
\[ \text{Area outside the rays} = \frac{1}{2} \times \text{Area of the sector of the larger circle} - \frac{1}{2} \times \text{Area of the sector of the smaller circle} \]
The angle between the rays can be calculated using their slopes, which are perpendicular, hence the angle is \(90^\circ\) or \(\frac{\pi}{2}\) radians.

Thus, the area of the sector for the larger circle is:
\[ \text{Sector area} = \frac{1}{4} \times 4\pi = \pi \]
And for the smaller circle:
\[ \text{Sector area} = \frac{1}{4} \times \pi = \frac{\pi}{4} \]

### Final Calculation
The area bounded by the circles and the rays is:
\[ \text{Total bounded area} = 3\pi - \left(\pi - \frac{\pi}{4}\right) = 3\pi - \frac{3\pi}{4} = \frac{12\pi}{4} - \frac{3\pi}{4} = \frac{9\pi}{4} \]

### Conclusion
The area bounded by the circles and the rays is:
\[ \frac{9\pi}{4} \]