If two circles touching both the axes intersect at two points P and Q where P=(3,1) then PQ=

Options

A. `sqrt(2)`
B. `2sqrt(2)`
C. `3sqrt(2)`
D. `4sqrt(2)`

Answer

To find the length of segment PQ where P = (3, 1) and Q = (1, 3), we can use the distance formula. The distance formula between two points (x1, y1) and (x2, y2) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step-by-Step Solution:

1. **Identify the coordinates of points P and Q**:
- P = (3, 1)
- Q = (1, 3)

2. **Substitute the coordinates into the distance formula**:
- Here, \( x_1 = 3 \), \( y_1 = 1 \), \( x_2 = 1 \), and \( y_2 = 3 \).
- The distance \( PQ \) can be calculated as follows:
\[ PQ = \sqrt{(1 - 3)^2 + (3 - 1)^2} \]

3. **Calculate the differences**:
- \( x_2 - x_1 = 1 - 3 = -2 \)
- \( y_2 - y_1 = 3 - 1 = 2 \)

4. **Square the differences**:
- \( (-2)^2 = 4 \)
- \( (2)^2 = 4 \)

5. **Add the squared differences**:
\[ PQ = \sqrt{4 + 4} = \sqrt{8} \]

6. **Simplify the square root**:
\[ PQ = \sqrt{8} = 2\sqrt{2} \]

### Final Answer:
The length of segment PQ is \( 2\sqrt{2} \).

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