The common chord of `x^(2)+y^(2)-4x-4y=0 and x^(2)+y^(2)=16` subtends at the origin an angle equal to

Options

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

Answer

To solve the problem, we need to find the angle subtended by the common chord of the two given circles at the origin. Let's go through the steps systematically.

### Step 1: Write the equations of the circles
The equations of the circles are:
1. \( x^2 + y^2 - 4x - 4y = 0 \) (Circle 1)
2. \( x^2 + y^2 = 16 \) (Circle 2)

### Step 2: Rearrange Circle 1 into standard form
We can rearrange Circle 1's equation into standard form by completing the square:
\[ x^2 - 4x + y^2 - 4y = 0 \]
Completing the square for \(x\) and \(y\):
\[ (x - 2)^2 - 4 + (y - 2)^2 - 4 = 0 \]
\[ (x - 2)^2 + (y - 2)^2 = 8 \]
This represents a circle with center at (2, 2) and radius \( \sqrt{8} = 2\sqrt{2} \).

### Step 3: Identify the second circle's center and radius
Circle 2 is already in standard form:
\[ x^2 + y^2 = 16 \]
This represents a circle with center at (0, 0) and radius \( \sqrt{16} = 4 \).

### Step 4: Find the equation of the common chord
To find the common chord, we subtract the equations of the two circles:
\[ (x^2 + y^2 - 4x - 4y) - (x^2 + y^2 - 16) = 0 \]
This simplifies to:
\[ -4x - 4y + 16 = 0 \]
Dividing through by -4 gives:
\[ x + y - 4 = 0 \]
or
\[ x + y = 4 \]

### Step 5: Determine the angle subtended at the origin
The line \(x + y = 4\) intersects the axes at:
- \(x\)-intercept: Set \(y = 0\) → \(x = 4\) → Point (4, 0)
- \(y\)-intercept: Set \(x = 0\) → \(y = 4\) → Point (0, 4)

### Step 6: Calculate the angle subtended at the origin
The angle subtended by the line segments from the origin to the points (4, 0) and (0, 4) can be calculated using the tangent of the angle:
- The slope of the line from the origin to (4, 0) is \(0\).
- The slope of the line from the origin to (0, 4) is undefined (vertical line).

The angle between these two lines is \(90^\circ\) or \(\frac{\pi}{2}\) radians.

### Final Answer
The angle subtended at the origin by the common chord is:
\[ \frac{\pi}{2} \text{ radians} \]