The common chord of the circles `x^(2)+y^(2)-4x-4y=0` and `2x^(2)+2y^(2)=32` subtends at the origin an angle equal to

Options

A. `pi/2`
B. `pi/4`
C. `pi/6`
D. `pi/3`

Answer

To solve the problem, we need to find the angle subtended by the common chord of the two circles at the origin. Let's break this down step by step.

### Step 1: Write the equations of the circles
The equations given are:
1. \( x^2 + y^2 - 4x - 4y = 0 \)
2. \( 2x^2 + 2y^2 = 32 \)

### Step 2: Simplify the second circle's equation
We can simplify the second equation by dividing everything by 2:
\[ x^2 + y^2 = 16 \]

### Step 3: Rewrite the first circle's equation
We can rearrange the first circle's equation:
\[ x^2 + y^2 - 4x - 4y = 0 \implies x^2 + y^2 = 4x + 4y \]

### Step 4: Find the common chord
The common chord can be found by subtracting the two equations:
\[ S_1 - S_2 = 0 \]
This gives us:
\[ (x^2 + y^2 - 4x - 4y) - (x^2 + y^2 - 16) = 0 \]
Simplifying this, we have:
\[ -4x - 4y + 16 = 0 \implies 4x + 4y = 16 \implies x + y = 4 \]

### Step 5: Find the points of intersection
To find the points of intersection of the common chord with the circles, we can substitute \( y = 4 - x \) into the equation of the second circle:
\[ x^2 + (4 - x)^2 = 16 \]
Expanding this:
\[ x^2 + (16 - 8x + x^2) = 16 \implies 2x^2 - 8x = 0 \]
Factoring out \( 2x \):
\[ 2x(x - 4) = 0 \]
This gives us \( x = 0 \) or \( x = 4 \). Correspondingly, the points are:
- For \( x = 0 \), \( y = 4 \) → Point A: \( (0, 4) \)
- For \( x = 4 \), \( y = 0 \) → Point B: \( (4, 0) \)

### Step 6: Calculate the angle subtended at the origin
The points A and B form a right triangle with the origin (0,0). The coordinates of the points are:
- A: \( (0, 4) \)
- B: \( (4, 0) \)

The angle \( \theta \) subtended at the origin by the line segments OA and OB can be found using the tangent function:
\[ \tan(\theta) = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{0 - 4} = -1 \]
This implies:
\[ \theta = \tan^{-1}(-1) = \frac{3\pi}{4} \]

However, since we are looking for the angle subtended by the chord, we need to find the angle between the lines OA and OB. The angle between the two lines can be calculated as:
\[ \theta = \pi - \frac{3\pi}{4} = \frac{\pi}{4} \]

### Conclusion
Thus, the angle subtended by the common chord at the origin is:
\[ \theta = \frac{\pi}{2} \]