Show that the angle between the circles
`x^2+y^2=a^2, x^2+y^2=ax+ay` is `(3pi)/4`

Answer

To show that the angle between the circles \( x^2 + y^2 = a^2 \) and \( x^2 + y^2 = ax + ay \) is \( \frac{3\pi}{4} \), we will follow these steps:

### Step 1: Identify the equations of the circles
The first circle \( S_1 \) is given by:
\[ x^2 + y^2 = a^2 \]
The second circle \( S_2 \) is given by:
\[ x^2 + y^2 = ax + ay \]

### Step 2: Rewrite the equation of the second circle
We can rewrite the equation of the second circle \( S_2 \) as:
\[ x^2 + y^2 - ax - ay = 0 \]
This can be rearranged to:
\[ x^2 - ax + y^2 - ay = 0 \]

### Step 3: Find the points of intersection
To find the points of intersection of the two circles, we can set the two equations equal to each other:
\[ a^2 = ax + ay \]
This simplifies to:
\[ ax + ay - a^2 = 0 \]
Factoring out \( a \):
\[ a(x + y - a) = 0 \]
Thus, we have two cases:
1. \( a = 0 \) (not applicable since \( a \) is a constant)
2. \( x + y = a \)

### Step 4: Substitute \( y = a - x \) into the first circle
Substituting \( y = a - x \) into the first circle's equation:
\[ x^2 + (a - x)^2 = a^2 \]
Expanding this:
\[ x^2 + (a^2 - 2ax + x^2) = a^2 \]
This simplifies to:
\[ 2x^2 - 2ax = 0 \]
Factoring gives:
\[ 2x(x - a) = 0 \]
Thus, \( x = 0 \) or \( x = a \).

### Step 5: Find corresponding \( y \) values
If \( x = 0 \), then \( y = a \).
If \( x = a \), then \( y = 0 \).
So the points of intersection are \( (0, a) \) and \( (a, 0) \).

### Step 6: Find the slopes of the tangents at the points of intersection
1. **For the first circle \( S_1 \)** at \( (0, a) \):
The slope \( m_1 \) can be found by implicit differentiation:
\[ 2x + 2yy' = 0 \implies y' = -\frac{x}{y} \]
At \( (0, a) \):
\[ m_1 = -\frac{0}{a} = 0 \]

2. **For the second circle \( S_2 \)** at \( (0, a) \):
The slope \( m_2 \):
\[ 2x + 2yy' - a - ay' = 0 \implies y'(2y - a) = a - 2x \]
At \( (0, a) \):
\[ y' = \frac{a - 0}{2a - a} = 1 \]

### Step 7: Calculate the angle between the tangents
Using the formula for the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \):
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Substituting \( m_1 = 0 \) and \( m_2 = 1 \):
\[ \tan \theta = \frac{0 - 1}{1 + 0 \cdot 1} = -1 \]
Thus, \( \theta = \tan^{-1}(-1) = \frac{3\pi}{4} \).

### Conclusion
The angle between the two circles is \( \frac{3\pi}{4} \).

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