The point `( [ P + 1 ] , [ P ] )` (where [.] denotes the greatest integer function), lying inside the region bounded by the circle `x^2 + y^2 - 2x - 15 = 0 and x^2 + y^2 - 2x - 7 =0,` then :

Options

A. a, `P""in[-1,0)uu[0,1)uu[1,2)`
B. b. `P""in[-1,2)-{0,-1}`
C. c. `P""in(-1,2)`
D. d. None of these

Answer

To solve the problem, we need to analyze the conditions given by the two circles and the point `( [P + 1], [P] )`, where `[.]` denotes the greatest integer function.

### Step 1: Rewrite the equations of the circles
The equations of the circles are given as:
1. \( x^2 + y^2 - 2x - 15 = 0 \)
2. \( x^2 + y^2 - 2x - 7 = 0 \)

We can rewrite these equations in standard form.

For the first circle:
\[ x^2 - 2x + y^2 = 15 \implies (x - 1)^2 + y^2 = 16 \]
This is a circle centered at \( (1, 0) \) with a radius of \( 4 \).

For the second circle:
\[ x^2 - 2x + y^2 = 7 \implies (x - 1)^2 + y^2 = 8 \]
This is a circle centered at \( (1, 0) \) with a radius of \( 2\sqrt{2} \).

### Step 2: Determine the conditions for the point to lie inside the first circle
The point \( ( [P + 1], [P] ) \) must satisfy the condition of lying inside the first circle. Therefore, we need:
\[ ([P + 1] - 1)^2 + [P]^2 < 16 \]
Let \( a = [P] \). Then, \( [P + 1] = a + 1 \). Substituting these values gives:
\[ (a + 1 - 1)^2 + a^2 < 16 \implies a^2 + a^2 < 16 \implies 2a^2 < 16 \implies a^2 < 8 \]
Thus, we have:
\[ a < 2\sqrt{2} \quad \text{(approximately 2.83)} \]

### Step 3: Determine the conditions for the point to lie outside the second circle
Next, we need the point to lie outside the second circle:
\[ ([P + 1] - 1)^2 + [P]^2 > 8 \]
Using the same substitution:
\[ (a + 1 - 1)^2 + a^2 > 8 \implies a^2 + a^2 > 8 \implies 2a^2 > 8 \implies a^2 > 4 \]
Thus, we have:
\[ a > 2 \]

### Step 4: Combine the inequalities
Now we combine the inequalities from both conditions:
1. \( a^2 < 8 \) implies \( -2\sqrt{2} < a < 2\sqrt{2} \)
2. \( a^2 > 4 \) implies \( a > 2 \) or \( a < -2 \)

Since \( a \) must be an integer (as it is the greatest integer function of \( P \)), we need to find integer values of \( a \) that satisfy:
\[ 2 < a < 2\sqrt{2} \quad \text{(approximately 2.83)} \]
The only integer value that satisfies this condition is \( a = 2 \).

### Step 5: Conclusion
Since \( a = [P] \) and \( [P] = 2 \), we conclude that:
\[ 2 \leq P < 3 \]
Thus, the possible values of \( P \) are in the interval \( [2, 3) \).

### Final Answer
The value of \( P \) lies in the interval \( [2, 3) \).