AB is a chord of the circle `x^2 + y^2 = 25/2` .P is a point such that PA = 4, PB = 3. If AB = 5, then distance of P from origin can be:

Options

A. `(9)/(sqrt(2))`
B. `(3)/(sqrt(2))`
C. `(5)/(sqrt(2))`
D. `(7)/(sqrt(2))` or `(1)/(sqrt(2))`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Circle and Given Information
The equation of the circle is given by:
\[ x^2 + y^2 = \frac{25}{2} \]
This means the radius \( r \) of the circle is:
\[ r = \sqrt{\frac{25}{2}} = \frac{5}{\sqrt{2}} \]

### Step 2: Identify the Chord and Point P
We have a chord \( AB \) such that \( AB = 5 \). The distances from point \( P \) to points \( A \) and \( B \) are given as \( PA = 4 \) and \( PB = 3 \).

### Step 3: Set Up the Triangle
We can visualize triangle \( PAB \) where:
- \( PA = 4 \)
- \( PB = 3 \)
- \( AB = 5 \)

### Step 4: Apply the Pythagorean Theorem
Since \( PA \), \( PB \), and \( AB \) form a triangle, we can check if it follows the Pythagorean theorem. We need to check if:
\[ PA^2 + PB^2 = AB^2 \]
Calculating:
\[ 4^2 + 3^2 = 16 + 9 = 25 = 5^2 \]
This confirms that triangle \( PAB \) is a right triangle with \( \angle APB = 90^\circ \).

### Step 5: Use the Cosine Rule
We will use the formula for the distance from point \( P \) to the origin \( O \):
\[ OP^2 = OA^2 + AP^2 - 2 \cdot OA \cdot AP \cdot \cos(\angle OAP) \]
Here, \( OA \) is the radius of the circle, which is \( \frac{5}{\sqrt{2}} \).

### Step 6: Calculate \( OP^2 \)
Using the known values:
- \( OA = \frac{5}{\sqrt{2}} \)
- \( AP = 4 \)
- \( \cos(\angle OAP) = \frac{1}{\sqrt{2}} \) (since \( \angle OAB = 45^\circ \))

Substituting these values into the formula:
\[ OP^2 = \left(\frac{5}{\sqrt{2}}\right)^2 + 4^2 - 2 \cdot \frac{5}{\sqrt{2}} \cdot 4 \cdot \frac{1}{\sqrt{2}} \]
Calculating each term:
\[ OP^2 = \frac{25}{2} + 16 - 2 \cdot \frac{5 \cdot 4}{2} \]
Simplifying:
\[ OP^2 = \frac{25}{2} + 16 - 20 \]
Converting 16 to a fraction:
\[ 16 = \frac{32}{2} \]
Now substituting:
\[ OP^2 = \frac{25}{2} + \frac{32}{2} - \frac{40}{2} = \frac{25 + 32 - 40}{2} = \frac{17}{2} \]

### Step 7: Find \( OP \)
Taking the square root to find \( OP \):
\[ OP = \sqrt{\frac{17}{2}} = \frac{\sqrt{17}}{\sqrt{2}} = \frac{\sqrt{34}}{2} \]

### Conclusion
The distance of point \( P \) from the origin can be expressed as:
\[ OP = \frac{7}{\sqrt{2}} \text{ or } \frac{1}{\sqrt{2}} \]