Find the coordinates of the centres and the radii of the circles whose equations are
`x^(2) + y^(2) = k (x + k)`

Answer

To find the coordinates of the center and the radius of the circle given by the equation \( x^2 + y^2 = k(x + k) \), we will follow these steps:

### Step 1: Rearrange the equation
Start by rewriting the given equation:
\[ x^2 + y^2 = kx + k^2 \]
We can rearrange it to bring all terms to one side:
\[ x^2 + y^2 - kx - k^2 = 0 \]

### Step 2: Identify coefficients
Now, we can compare this equation with the standard form of a circle:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From our equation, we can identify:
- The coefficient of \( x \) is \( -k \) (which means \( 2g = -k \))
- The coefficient of \( y \) is \( 0 \) (which means \( 2f = 0 \))
- The constant term is \( -k^2 \) (which means \( c = -k^2 \))

### Step 3: Find the center coordinates
Using the relationships from the standard form:
- The x-coordinate of the center is given by \( -\frac{2g}{2} = -g \)
- The y-coordinate of the center is given by \( -\frac{2f}{2} = -f \)

Substituting the values of \( g \) and \( f \):
- \( g = -\frac{k}{2} \) so \( -g = \frac{k}{2} \)
- \( f = 0 \) so \( -f = 0 \)

Thus, the coordinates of the center are:
\[ \left( \frac{k}{2}, 0 \right) \]

### Step 4: Find the radius
The radius \( r \) of the circle can be calculated using the formula:
\[ r = \sqrt{g^2 + f^2 - c} \]
Substituting the values we found:
- \( g = -\frac{k}{2} \)
- \( f = 0 \)
- \( c = -k^2 \)

Calculating \( r \):
\[ r = \sqrt{\left(-\frac{k}{2}\right)^2 + 0^2 - (-k^2)} \]
\[ r = \sqrt{\frac{k^2}{4} + k^2} \]
\[ r = \sqrt{\frac{k^2}{4} + \frac{4k^2}{4}} = \sqrt{\frac{5k^2}{4}} = \frac{\sqrt{5}k}{2} \]

### Final Result
The coordinates of the center and the radius of the circle are:
- Center: \( \left( \frac{k}{2}, 0 \right) \)
- Radius: \( \frac{\sqrt{5}k}{2} \)

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