Find the equation to the circle
Whose radius is a + b and whose centre is (a, - b).

Answer

To find the equation of the circle with a given center and radius, we can follow these steps:

### Step 1: Identify the center and radius
The center of the circle is given as \( (a, -b) \) and the radius is \( r = a + b \).

### Step 2: Write the standard form of the circle's equation
The standard equation of a circle with center \( (h, k) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

### Step 3: Substitute the center and radius into the equation
Here, \( h = a \), \( k = -b \), and \( r = a + b \). Substituting these values into the standard equation gives:
\[ (x - a)^2 + (y + b)^2 = (a + b)^2 \]

### Step 4: Expand both sides of the equation
Now, we will expand both sides of the equation:
1. Left side:
\[ (x - a)^2 = x^2 - 2ax + a^2 \]
\[ (y + b)^2 = y^2 + 2by + b^2 \]
So, the left side becomes:
\[ x^2 - 2ax + a^2 + y^2 + 2by + b^2 \]

2. Right side:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]

### Step 5: Set the equation
Now we equate both sides:
\[ x^2 - 2ax + a^2 + y^2 + 2by + b^2 = a^2 + 2ab + b^2 \]

### Step 6: Simplify the equation
Now, we can simplify this equation:
- The \( a^2 \) and \( b^2 \) terms on both sides cancel out:
\[ x^2 + y^2 - 2ax + 2by = 2ab \]

### Step 7: Rearrange to standard form
Rearranging gives us:
\[ x^2 + y^2 - 2ax + 2by - 2ab = 0 \]

### Final Equation
Thus, the equation of the circle is:
\[ x^2 + y^2 - 2ax + 2by - 2ab = 0 \]

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