If the length of the tangent from `(2,5)` to
the circle `x^(2) + y^(2) - 5x +4y + k = 0` is
`sqrt(37) ` then find k.

Answer

To solve the problem, we need to find the value of \( k \) such that the length of the tangent from the point \( (2, 5) \) to the circle defined by the equation \( x^2 + y^2 - 5x + 4y + k = 0 \) is equal to \( \sqrt{37} \).

### Step-by-step Solution:

1. **Identify the Circle Equation**:
The given equation of the circle is:
\[ x^2 + y^2 - 5x + 4y + k = 0 \]
This can be rearranged to:
\[ x^2 + y^2 - 5x + 4y = -k \]

2. **Convert to Standard Form**:
To find the center and radius of the circle, we need to complete the square for \( x \) and \( y \).

- For \( x^2 - 5x \):
\[ x^2 - 5x = (x - \frac{5}{2})^2 - \frac{25}{4} \]
- For \( y^2 + 4y \):
\[ y^2 + 4y = (y + 2)^2 - 4 \]

Substituting these back into the equation gives:
\[ (x - \frac{5}{2})^2 - \frac{25}{4} + (y + 2)^2 - 4 = -k \]
Simplifying this, we get:
\[ (x - \frac{5}{2})^2 + (y + 2)^2 = -k + \frac{25}{4} + 4 \]
\[ (x - \frac{5}{2})^2 + (y + 2)^2 = -k + \frac{25}{4} + \frac{16}{4} \]
\[ (x - \frac{5}{2})^2 + (y + 2)^2 = -k + \frac{41}{4} \]

3. **Identify the Center and Radius**:
The center of the circle is \( \left(\frac{5}{2}, -2\right) \) and the radius \( r \) is given by:
\[ r = \sqrt{-k + \frac{41}{4}} \]

4. **Length of the Tangent Formula**:
The length of the tangent from a point \( (x_1, y_1) \) to the circle is given by:
\[ L = \sqrt{(x_1 - h)^2 + (y_1 - k)^2 - r^2} \]
where \( (h, k) \) is the center of the circle. Here, \( (x_1, y_1) = (2, 5) \).

5. **Calculate the Length of the Tangent**:
Substituting the values:
\[ L = \sqrt{\left(2 - \frac{5}{2}\right)^2 + (5 + 2)^2 - \left(-k + \frac{41}{4}\right)} \]
Simplifying:
\[ L = \sqrt{\left(-\frac{1}{2}\right)^2 + 7^2 - \left(-k + \frac{41}{4}\right)} \]
\[ L = \sqrt{\frac{1}{4} + 49 - (-k + \frac{41}{4})} \]
\[ L = \sqrt{\frac{1}{4} + 49 + k - \frac{41}{4}} \]
\[ L = \sqrt{49 - 10 + k} = \sqrt{39 + k} \]

6. **Set the Length Equal to \( \sqrt{37} \)**:
According to the problem, this length is equal to \( \sqrt{37} \):
\[ \sqrt{39 + k} = \sqrt{37} \]

7. **Square Both Sides**:
Squaring both sides gives:
\[ 39 + k = 37 \]

8. **Solve for \( k \)**:
Rearranging gives:
\[ k = 37 - 39 = -2 \]

### Final Answer:
Thus, the value of \( k \) is:
\[ \boxed{-2} \]