The radius of the circle `x^(2) + y^(2) -2x + 3y+k = 0` is 2 `1/2` Find the value of k. Find also the equation of the diameter of the circle, which passes through the point `(5,2 (1)/(2))`

Answer

To solve the problem step by step, we need to follow these steps:

### Step 1: Identify the standard form of the circle equation
The given equation of the circle is:
\[ x^2 + y^2 - 2x + 3y + k = 0 \]

We can rewrite it in the standard form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center of the circle and \(r\) is the radius.

### Step 2: Rearrange the equation
Rearranging the given equation, we have:
\[ x^2 - 2x + y^2 + 3y + k = 0 \]

### Step 3: Complete the square for \(x\) and \(y\)
1. For \(x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]
2. For \(y\):
\[ y^2 + 3y = (y + \frac{3}{2})^2 - \frac{9}{4} \]

Substituting these back into the equation gives:
\[ (x - 1)^2 - 1 + (y + \frac{3}{2})^2 - \frac{9}{4} + k = 0 \]

### Step 4: Simplify the equation
Combining the constants:
\[ (x - 1)^2 + (y + \frac{3}{2})^2 + k - 1 - \frac{9}{4} = 0 \]
\[ (x - 1)^2 + (y + \frac{3}{2})^2 + k - \frac{13}{4} = 0 \]

### Step 5: Set the equation equal to the radius squared
Since the radius is given as \(2 \frac{1}{2} = \frac{5}{2}\), we have:
\[ r^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]

Thus, we set:
\[ k - \frac{13}{4} = -\frac{25}{4} \]

### Step 6: Solve for \(k\)
Rearranging gives:
\[ k = -\frac{25}{4} + \frac{13}{4} \]
\[ k = -\frac{12}{4} \]
\[ k = -3 \]

### Step 7: Find the center of the circle
From the completed square form, the center of the circle is:
\[ (h, k) = (1, -\frac{3}{2}) \]

### Step 8: Find the equation of the diameter
The diameter passes through the center \((1, -\frac{3}{2})\) and the point \((5, 2.5)\).

### Step 9: Calculate the slope of the diameter
The slope \(m\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2.5 - (-\frac{3}{2})}{5 - 1} \]
\[ = \frac{2.5 + 1.5}{4} = \frac{4}{4} = 1 \]

### Step 10: Use point-slope form to find the equation of the diameter
Using the point-slope form:
\[ y - y_1 = m(x - x_1) \]
Substituting \((x_1, y_1) = (1, -\frac{3}{2})\) and \(m = 1\):
\[ y + \frac{3}{2} = 1(x - 1) \]
\[ y + \frac{3}{2} = x - 1 \]
\[ x - y - \frac{5}{2} = 0 \]

Thus, the final equation of the diameter is:
\[ x - y - 2.5 = 0 \]

### Final Answers
- The value of \(k\) is \(-3\).
- The equation of the diameter is \(x - y - 2.5 = 0\).