If the distance between the points (a, 2, 1) and (1, -1, 1) is 5, then value of a are :

Options

A. `-5, -3`
B. `5, -3`
C. `-5, 3`
D. 5, 3

Answer

To solve the problem of finding the value of \( a \) such that the distance between the points \( (a, 2, 1) \) and \( (1, -1, 1) \) is 5, we can follow these steps:

### Step 1: Use the Distance Formula
The distance \( d \) between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in 3D space is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 2: Identify the Points
Here, the points are:
- Point 1: \( (a, 2, 1) \) which gives us \( x_1 = a, y_1 = 2, z_1 = 1 \)
- Point 2: \( (1, -1, 1) \) which gives us \( x_2 = 1, y_2 = -1, z_2 = 1 \)

### Step 3: Substitute into the Distance Formula
Substituting the coordinates into the distance formula:
\[ d = \sqrt{(1 - a)^2 + (-1 - 2)^2 + (1 - 1)^2} \]
This simplifies to:
\[ d = \sqrt{(1 - a)^2 + (-3)^2 + 0^2} \]
\[ d = \sqrt{(1 - a)^2 + 9} \]

### Step 4: Set the Distance Equal to 5
According to the problem, the distance is 5:
\[ \sqrt{(1 - a)^2 + 9} = 5 \]

### Step 5: Square Both Sides
To eliminate the square root, we square both sides:
\[ (1 - a)^2 + 9 = 25 \]

### Step 6: Simplify the Equation
Now, we simplify the equation:
\[ (1 - a)^2 = 25 - 9 \]
\[ (1 - a)^2 = 16 \]

### Step 7: Take the Square Root
Taking the square root of both sides gives us two equations:
\[ 1 - a = 4 \quad \text{or} \quad 1 - a = -4 \]

### Step 8: Solve for \( a \)
1. From \( 1 - a = 4 \):
\[ -a = 4 - 1 \implies -a = 3 \implies a = -3 \]

2. From \( 1 - a = -4 \):
\[ -a = -4 - 1 \implies -a = -5 \implies a = 5 \]

### Step 9: Final Values of \( a \)
Thus, the values of \( a \) are:
\[ a = -3 \quad \text{and} \quad a = 5 \]

### Conclusion
The final answer is:
\[ \text{The values of } a \text{ are } -3 \text{ and } 5. \]