The distance between two points in space , `,P(x,-1,-1)and Q(3,-3,1)` , is 3 .find the possible values of x.

Options

A. 1 or 2
B. 2 or 3
C. `-2 or 3 `
D. 2 or 4

Explanation

To find the possible values of \( x \) such that the distance between the points \( P(x, -1, -1) \) and \( Q(3, -3, 1) \) is 3, we can follow these steps:

### Step 1: Write down the distance formula
The distance \( D \) between two points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) in three-dimensional space is given by:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 2: Substitute the coordinates of points P and Q
Here, the coordinates of point \( P \) are \( (x, -1, -1) \) and those of point \( Q \) are \( (3, -3, 1) \). Substituting these coordinates into the distance formula gives:
\[ 3 = \sqrt{(3 - x)^2 + (-3 - (-1))^2 + (1 - (-1))^2} \]

### Step 3: Simplify the equation
Now, simplify the terms inside the square root:
- For the \( y \)-coordinates:
\[ -3 - (-1) = -3 + 1 = -2 \quad \Rightarrow \quad (-2)^2 = 4 \]
- For the \( z \)-coordinates:
\[ 1 - (-1) = 1 + 1 = 2 \quad \Rightarrow \quad (2)^2 = 4 \]
Now substituting these back into the equation:
\[ 3 = \sqrt{(3 - x)^2 + 4 + 4} \]
This simplifies to:
\[ 3 = \sqrt{(3 - x)^2 + 8} \]

### Step 4: Square both sides
To eliminate the square root, square both sides:
\[ 3^2 = (3 - x)^2 + 8 \]
This gives:
\[ 9 = (3 - x)^2 + 8 \]

### Step 5: Isolate the squared term
Now, isolate the squared term:
\[ (3 - x)^2 = 9 - 8 \]
\[ (3 - x)^2 = 1 \]

### Step 6: Take the square root
Taking the square root of both sides gives:
\[ 3 - x = \pm 1 \]

### Step 7: Solve for \( x \)
Now, we can solve for \( x \) in two cases:
1. \( 3 - x = 1 \)
\[ x = 3 - 1 = 2 \]
2. \( 3 - x = -1 \)
\[ x = 3 + 1 = 4 \]

### Conclusion
The possible values of \( x \) are:
\[ x = 2 \quad \text{or} \quad x = 4 \]