Let P and Q are two points on the curve `y=log_((1)/(2))(x-0.5)+log_2sqrt(4x^(2)-4x+1)` and P is also on the circle `x^(2)+y^(2)=10`. Q lies inside the given circle such that its abscissa is an integer.
Q. `OP*OQ`, O being the origin is

Options

A. 4 or 7
B. 4 or 2
C. 2 or 3
D. 7 or 8

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and derive the necessary calculations.

### Step 1: Understand the curve equation
The curve is given by:
\[ y = \log_{(1/2)}(x - 0.5) + \log_{2}(\sqrt{4x^2 - 4x + 1}) \]

### Step 2: Simplify the curve equation
Using properties of logarithms, we can rewrite the equation:
\[ \log_{(1/2)}(x - 0.5) = -\log_{2}(x - 0.5) \]
Thus,
\[ y = -\log_{2}(x - 0.5) + \log_{2}(\sqrt{4x^2 - 4x + 1}) \]
Since \(\sqrt{4x^2 - 4x + 1} = \sqrt{(2x - 1)^2} = |2x - 1|\), we can write:
\[ y = -\log_{2}(x - 0.5) + \log_{2}(|2x - 1|) \]

### Step 3: Further simplify
Using the logarithmic identity \(\log_{a}(b) - \log_{a}(c) = \log_{a}(\frac{b}{c})\):
\[ y = \log_{2}\left(\frac{|2x - 1|}{x - 0.5}\right) \]

### Step 4: Find point P on the curve that also lies on the circle
The circle is given by:
\[ x^2 + y^2 = 10 \]
Let \( P = (x_1, y_1) \). Since \( y_1 = 1 \) (as derived from the curve), we substitute into the circle equation:
\[ x_1^2 + 1^2 = 10 \]
\[ x_1^2 + 1 = 10 \]
\[ x_1^2 = 9 \]
\[ x_1 = 3 \text{ or } -3 \]

### Step 5: Check which x value is valid
For \( x_1 = -3 \):
\[ y = \log_{(1/2)}(-3 - 0.5) + \log_{2}(\sqrt{4(-3)^2 - 4(-3) + 1}) \]
This would yield a negative argument for the logarithm, which is invalid.

Thus, \( x_1 = 3 \) is valid, giving us:
\[ P = (3, 1) \]

### Step 6: Determine point Q
Point \( Q \) lies inside the circle and has an integer abscissa. The range for \( x \) is between \( 0.5 \) and \( 3 \) (not including \( 3 \)):
Possible integer values for \( x \) are \( 1 \) and \( 2 \).

### Step 7: Calculate coordinates for Q
1. For \( x_2 = 1 \):
\[ y = 1 \] (from the curve)
Thus, \( Q = (1, 1) \).

2. For \( x_2 = 2 \):
\[ y = 1 \] (from the curve)
Thus, \( Q = (2, 1) \).

### Step 8: Calculate the dot product \( OP \cdot OQ \)
1. For \( Q = (1, 1) \):
\[ OP = (3, 1) \]
\[ OQ = (1, 1) \]
\[ OP \cdot OQ = 3 \cdot 1 + 1 \cdot 1 = 3 + 1 = 4 \]

2. For \( Q = (2, 1) \):
\[ OP = (3, 1) \]
\[ OQ = (2, 1) \]
\[ OP \cdot OQ = 3 \cdot 2 + 1 \cdot 1 = 6 + 1 = 7 \]

### Final Answer
The values of \( OP \cdot OQ \) are \( 4 \) and \( 7 \).

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