Find the equation of the circle with centre (-a, -b) and radius `sqrt(a^(2) - b^(2))` .

Answer

To find the equation of the circle with center (-a, -b) and radius \( \sqrt{a^2 - b^2} \), we can follow these steps:

### Step 1: Write the standard equation of a circle
The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, \(h = -a\) and \(k = -b\), and the radius \(r = \sqrt{a^2 - b^2}\).

### Step 2: Substitute the center and radius into the equation
Substituting \(h\), \(k\), and \(r\) into the standard equation, we have:
\[ (x - (-a))^2 + (y - (-b))^2 = (\sqrt{a^2 - b^2})^2 \]
This simplifies to:
\[ (x + a)^2 + (y + b)^2 = a^2 - b^2 \]

### Step 3: Expand the left-hand side
Now, we will expand the left-hand side:
\[ (x + a)^2 + (y + b)^2 = (x^2 + 2ax + a^2) + (y^2 + 2by + b^2) \]
Combining these, we get:
\[ x^2 + 2ax + a^2 + y^2 + 2by + b^2 \]

### Step 4: Write the equation
Now, we equate the expanded left-hand side to the right-hand side:
\[ x^2 + 2ax + y^2 + 2by + a^2 + b^2 = a^2 - b^2 \]

### Step 5: Simplify the equation
Next, we can simplify the equation by moving all terms to one side:
\[ x^2 + 2ax + y^2 + 2by + a^2 + b^2 - a^2 + b^2 = 0 \]
This simplifies to:
\[ x^2 + 2ax + y^2 + 2by + 2b^2 = 0 \]

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

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