Find the angle between the pair of
tangents drawn from `(1, 3)` to the circle
`x^(2) + y^(2) - 2 x + 4y - 11 = 0`

Answer

To find the angle between the pair of tangents drawn from the point (1, 3) to the circle given by the equation \(x^2 + y^2 - 2x + 4y - 11 = 0\), we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we need to rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - 2x + 4y - 11 = 0 \]
We can rearrange it as:
\[ x^2 - 2x + y^2 + 4y = 11 \]

### Step 2: Complete the Square
Next, we complete the square for both \(x\) and \(y\):
- For \(x^2 - 2x\), we add and subtract \(1\) (which is \((\frac{-2}{2})^2\)):
\[ x^2 - 2x + 1 = (x - 1)^2 \]
- For \(y^2 + 4y\), we add and subtract \(4\) (which is \((\frac{4}{2})^2\)):
\[ y^2 + 4y + 4 = (y + 2)^2 \]

Now substituting back, we have:
\[ (x - 1)^2 + (y + 2)^2 = 16 \]

### Step 3: Identify the Center and Radius
From the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can identify:
- Center \((h, k) = (1, -2)\)
- Radius \(r = \sqrt{16} = 4\)

### Step 4: Calculate the Distance from the Point to the Center
Now, we calculate the distance \(d\) from the point \(P(1, 3)\) to the center \(C(1, -2)\):
\[ d = \sqrt{(1 - 1)^2 + (3 - (-2))^2} = \sqrt{0 + (3 + 2)^2} = \sqrt{5^2} = 5 \]

### Step 5: Use the Right Triangle to Find the Angle
We have a right triangle formed by the center \(C\), the point \(P\), and the point where the tangent touches the circle. The sides of the triangle are:
- The radius \(r = 4\)
- The distance from the point to the center \(d = 5\)
- The length of the tangent \(t\) can be calculated using the Pythagorean theorem:
\[ t^2 + r^2 = d^2 \implies t^2 + 4^2 = 5^2 \implies t^2 + 16 = 25 \implies t^2 = 9 \implies t = 3 \]

### Step 6: Calculate the Angle
Let \(\theta\) be the angle between the radius and the tangent. We can use the sine function:
\[ \sin \theta = \frac{r}{d} = \frac{4}{5} \]
Thus,
\[ \theta = \sin^{-1}\left(\frac{4}{5}\right) \]

### Step 7: Find the Angle Between the Tangents
The angle between the two tangents is \(2\theta\):
\[ \text{Angle between tangents} = 2\theta = 2 \sin^{-1}\left(\frac{4}{5}\right) \]

### Final Answer
The angle between the pair of tangents drawn from the point (1, 3) to the circle is:
\[ 2 \sin^{-1}\left(\frac{4}{5}\right) \]