The common chord of `x^(2)+y^(2)-4x-4y=0` and `x^(2)+y^2=4^(2)` subtends and angle `alpha` at the origin, then `alpha` equals

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

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

### Step 1: Identify the equations of the circles
The first circle is given by:
\[ x^2 + y^2 - 4x - 4y = 0 \]
The second circle is given by:
\[ x^2 + y^2 = 4^2 \]

### Step 2: Rewrite the first circle in standard form
We can rewrite the first circle's equation by completing the square:
\[ x^2 - 4x + y^2 - 4y = 0 \]
Completing the square for \(x\) and \(y\):
\[ (x - 2)^2 + (y - 2)^2 - 4 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 2)^2 = 4 \]
This indicates that the first circle has a center at \( (2, 2) \) and a radius of \( 2 \).

### Step 3: Identify the second circle's center and radius
The second circle can be rewritten as:
\[ x^2 + y^2 = 16 \]
This indicates that the second circle has a center at \( (0, 0) \) and a radius of \( 4 \).

### Step 4: Find the equation of the common chord
The common chord of the two circles can be found using the formula:
\[ S_1 - S_2 = 0 \]
Where \( S_1 \) is the equation of the first circle and \( S_2 \) is the equation of the second circle.

Substituting the equations:
\[ (x^2 + y^2 - 4x - 4y) - (x^2 + y^2 - 16) = 0 \]
This simplifies to:
\[ -4x - 4y + 16 = 0 \]
Rearranging gives:
\[ x + y = 4 \]

### Step 5: Determine the angle subtended at the origin
The line \( x + y = 4 \) intersects the axes at points \( (4, 0) \) and \( (0, 4) \). To find the angle \( \alpha \) subtended at the origin by this line, we can use the slopes of the line segments from the origin to these points.

The slope of the line from the origin to \( (4, 0) \) is:
\[ m_1 = \frac{0 - 0}{4 - 0} = 0 \]
The slope of the line from the origin to \( (0, 4) \) is:
\[ m_2 = \frac{4 - 0}{0 - 0} = \infty \]
The angle \( \alpha \) between these two lines can be calculated using the tangent of the angle between two lines:
\[ \tan(\alpha) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
However, since one slope is \( 0 \) and the other is undefined, we can see that the lines are perpendicular to each other. Thus, the angle \( \alpha \) is:
\[ \alpha = 90^\circ \]

### Final Answer
The angle \( \alpha \) subtended at the origin by the common chord is:
\[ \alpha = 90^\circ \]