Tangents PA and PB are drawn to the circle `(x-4)^(2)+(y-5)^(2)=4` from the point P on the curve`y=sin x` , where A and B lie on the circle. Consider the function `y= f(x)` represented by the locus of the center of the circumcircle of triangle PAB. Then answer the following questions.
Which of the following is true ?

Options

A. `f(x)=4` has real roots.
B. `f(x)=1 ` has real roots.
C. The range of `y=f^(-1)` is `[-(pi)/(4)+2,(pi)/(4)+2]`
D. None of these

Explanation

To solve the problem step-by-step, we need to analyze the given circle, the point P, and the tangents PA and PB. We will then derive the function \( y = f(x) \) that represents the locus of the center of the circumcircle of triangle PAB.

### Step 1: Identify the Circle and its Properties
The equation of the circle is given by:
\[ (x - 4)^2 + (y - 5)^2 = 4 \]
This represents a circle with center \( C(4, 5) \) and radius \( r = 2 \).

**Hint:** The center of the circle can be found directly from the equation of the circle.

### Step 2: Determine the Point P
The point P lies on the curve \( y = \sin x \). Thus, we can express the coordinates of point P as:
\[ P(t) = (t, \sin t) \]
where \( t \) is a parameter representing the x-coordinate.

**Hint:** The sine function oscillates between -1 and 1, so consider the range of \( y \) values for \( P(t) \).

### Step 3: Find the Tangents from Point P to the Circle
The tangents PA and PB from point P to the circle can be derived using the formula for the length of the tangent from a point to a circle. The length \( L \) of the tangent from point \( P(t) \) 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 and \( r \) is the radius.

Substituting the values:
\[ L = \sqrt{(t - 4)^2 + (\sin t - 5)^2 - 4} \]

**Hint:** Ensure that the distance calculated is non-negative for the tangents to exist.

### Step 4: Determine the Coordinates of the Center of the Circumcircle
The center \( D \) of the circumcircle of triangle PAB can be found as the midpoint of the tangents. The coordinates of \( D \) can be expressed as:
\[ D = \left( \frac{t + 4}{2}, \frac{\sin t + 5}{2} \right) \]

**Hint:** The circumcenter is the average of the coordinates of the points forming the triangle.

### Step 5: Express the Function \( y = f(x) \)
From the coordinates of \( D \), we can express the function \( y = f(x) \):
\[ f(x) = \frac{\sin(2x - 4) + 5}{2} \]

**Hint:** The function \( f(x) \) represents the locus of the circumcenter.

### Step 6: Analyze the Function for Roots and Range
To find the number of real roots and the range of \( f(x) \):
1. Set \( f(x) = 4 \) and solve for \( x \):
\[ \sin(2x - 4) = 3 \]
This has no real solutions since the sine function ranges from -1 to 1.

2. Set \( f(x) = 1 \) and solve for \( x \):
\[ \sin(2x - 4) = -1 \]
This also has no real solutions.

3. Determine the range of \( f^{-1}(x) \):
The range of \( f(x) \) can be derived from the sine function's properties, leading to:
\[ \text{Range of } f^{-1}(x) = \left(-\frac{5}{4} + 2, \frac{\pi}{4} + 2\right) \]

**Hint:** Use the properties of the sine function and its inverse to determine the range.

### Conclusion
After analyzing the options given in the question, we find that the third option corresponds to the range derived from our function \( f(x) \).

Thus, the correct answer is:
**Option 3 is correct.**