Equation of circles which touch both the axes and also the line `x=k(kgt0)` is

Options

A. `x^(2)+y^(2)-kx+-ky+(k^(2))/4=0`
B. `x^(2)+y^(2)+kx+-ky+(k^(2))/4=0`
C. `x^(2)+y^(2)+-kx+ky+(k^(2))/4=0`
D. `x^(2)+y^(2)+-kx-ky+(k^(2))/4=0`

Answer

To find the equations of circles that touch both the axes and also the line \( x = k \) (where \( k > 0 \)), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Circle's Properties**:
- A circle that touches both the x-axis and y-axis must have its center at \( (r, r) \), where \( r \) is the radius of the circle. This is because the distance from the center to both axes must equal the radius.

2. **Radius Determination**:
- Since the circle also touches the line \( x = k \), the distance from the center \( (r, r) \) to the line \( x = k \) must also equal the radius \( r \). The distance from a point \( (x_0, y_0) \) to a vertical line \( x = a \) is given by \( |x_0 - a| \).
- Therefore, the distance from the center \( (r, r) \) to the line \( x = k \) is \( |r - k| \).

3. **Setting Up the Equation**:
- Since the circle touches the line \( x = k \), we have:
\[ r = k - r \]
- Solving for \( r \):
\[ 2r = k \implies r = \frac{k}{2} \]

4. **Finding the Center**:
- The center of the circle is \( \left(\frac{k}{2}, \frac{k}{2}\right) \).

5. **Writing the Circle's Equation**:
- The standard form of a circle's equation is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
- Substituting \( h = \frac{k}{2} \), \( k = \frac{k}{2} \), and \( r = \frac{k}{2} \):
\[ \left(x - \frac{k}{2}\right)^2 + \left(y - \frac{k}{2}\right)^2 = \left(\frac{k}{2}\right)^2 \]

6. **Expanding the Equation**:
- Expanding the left-hand side:
\[ \left(x - \frac{k}{2}\right)^2 + \left(y - \frac{k}{2}\right)^2 = \frac{k^2}{4} \]
- This expands to:
\[ x^2 - kx + \frac{k^2}{4} + y^2 - ky + \frac{k^2}{4} = \frac{k^2}{4} \]
- Simplifying gives:
\[ x^2 + y^2 - kx - ky + \frac{k^2}{2} - \frac{k^2}{4} = 0 \]
- Thus, we have:
\[ x^2 + y^2 - kx - ky + \frac{k^2}{4} = 0 \]

7. **Considering the Second Circle**:
- For the circle that touches the x-axis and the line \( x = k \) but is below the x-axis, the center will be \( \left(\frac{k}{2}, -\frac{k}{2}\right) \) with the same radius \( \frac{k}{2} \).
- The equation will be:
\[ \left(x - \frac{k}{2}\right)^2 + \left(y + \frac{k}{2}\right)^2 = \left(\frac{k}{2}\right)^2 \]
- Following similar steps as above will yield:
\[ x^2 + y^2 - kx + ky + \frac{k^2}{4} = 0 \]

8. **Final Result**:
- The equations of the circles that touch both axes and the line \( x = k \) are:
\[ x^2 + y^2 - kx \pm ky + \frac{k^2}{4} = 0 \]