The polar coordinate of a point P is `(2, -(pi)/(4))`. The polar coordinate of the point Q, which is such that the line joining Pqbisected perpendicularly by the initial line, is

Options

A. `(2, (pi)/(4))`
B. `(2, (pi)/(6))`
C. `(-2, (pi)/(4))`
D. `(-2, (pi)/(6))`

Answer

To find the polar coordinates of point Q, we start with the given polar coordinates of point P, which are \( (2, -\frac{\pi}{4}) \).

### Step-by-step Solution:

1. **Understanding Polar Coordinates**:
- The polar coordinates of a point are given in the form \( (r, \theta) \), where \( r \) is the distance from the origin and \( \theta \) is the angle measured from the positive x-axis.
- For point P, \( r = 2 \) and \( \theta = -\frac{\pi}{4} \).

2. **Finding the Cartesian Coordinates of Point P**:
- The Cartesian coordinates \( (x, y) \) can be calculated using the formulas:
\[ x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta) \]
- Substituting the values for point P:
\[ x_P = 2 \cos\left(-\frac{\pi}{4}\right) = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2} \]
\[ y_P = 2 \sin\left(-\frac{\pi}{4}\right) = 2 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\sqrt{2} \]
- Thus, the Cartesian coordinates of point P are \( P(\sqrt{2}, -\sqrt{2}) \).

3. **Understanding the Condition for Point Q**:
- The problem states that the line joining points P and Q is bisected perpendicularly by the initial line (the positive x-axis).
- This means that the angle of point Q must be such that the angle between the line joining P and Q and the x-axis is \( 90^\circ \).

4. **Finding the Angle for Point Q**:
- Since the line joining P and Q is bisected perpendicularly by the x-axis, point Q must have an angle \( \theta_Q \) that is the opposite of \( \theta_P \).
- The angle \( \theta_P = -\frac{\pi}{4} \), so the angle for point Q will be:
\[ \theta_Q = \frac{\pi}{4} \]

5. **Finding the Distance for Point Q**:
- The distance \( r_Q \) for point Q is the same as that for point P, which is \( r_Q = 2 \).

6. **Final Polar Coordinates of Point Q**:
- Therefore, the polar coordinates of point Q are:
\[ Q(2, \frac{\pi}{4}) \]

### Final Answer:
The polar coordinates of point Q are \( (2, \frac{\pi}{4}) \).