Find the angle between the pair of
tangents drawn from `(0,0)` to the circle
`x^(2) + y^(2) - 14 x + 2y + 25 = 0.`

Answer

To find the angle between the pair of tangents drawn from the point (0,0) to the circle given by the equation \( x^2 + y^2 - 14x + 2y + 25 = 0 \), we will follow these steps:

### Step 1: Rewrite the Circle Equation
First, we need to rewrite the circle's equation in standard form. The given equation is:
\[ x^2 + y^2 - 14x + 2y + 25 = 0 \]
We can complete the square for the \(x\) and \(y\) terms.

### Step 2: Completing the Square
For \(x\):
\[ x^2 - 14x \rightarrow (x - 7)^2 - 49 \]
For \(y\):
\[ y^2 + 2y \rightarrow (y + 1)^2 - 1 \]
Substituting these back into the equation gives:
\[ (x - 7)^2 - 49 + (y + 1)^2 - 1 + 25 = 0 \]
Simplifying this:
\[ (x - 7)^2 + (y + 1)^2 - 25 = 0 \]
Thus, we have:
\[ (x - 7)^2 + (y + 1)^2 = 25 \]
This represents a circle with center \(C(7, -1)\) and radius \(r = 5\).

### Step 3: Calculate the Distance from the Point to the Center
Next, we find the distance \(OC\) from the origin (0,0) to the center \(C(7, -1)\):
\[ OC = \sqrt{(7 - 0)^2 + (-1 - 0)^2} = \sqrt{7^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2} \]

### Step 4: Use the Tangent Angle Formula
The angle \( \theta \) between the tangents from a point to a circle can be found using the formula:
\[ \tan(\theta) = \frac{r}{d} \]
where \(r\) is the radius and \(d\) is the distance from the point to the center of the circle.

Here, \(r = 5\) and \(d = 5\sqrt{2}\):
\[ \tan(\theta) = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 5: Find the Angle \( \theta \)
From the tangent value, we can find \( \theta \):
\[ \theta = 45^\circ \quad \text{(since } \tan(45^\circ) = 1) \]

### Step 6: Calculate the Angle Between the Tangents
The angle between the two tangents is \(2\theta\):
\[ 2\theta = 2 \times 45^\circ = 90^\circ \]

### Final Answer
Thus, the angle between the pair of tangents drawn from the origin to the circle is:
\[ \boxed{90^\circ} \]