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

Options

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

Answer

To find the value(s) of \( a \) such that the distance between the points \( (a, 2, 1) \) and \( (1, -1, 1) \) is equal to 5, we can use the distance formula in three-dimensional space.

### Step-by-Step Solution:

1. **Distance Formula**: The distance \( d \) between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in three-dimensional space is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

2. **Assign Points**: Here, we have the points \( (a, 2, 1) \) and \( (1, -1, 1) \). Thus, we can assign:
- \( x_1 = a \), \( y_1 = 2 \), \( z_1 = 1 \)
- \( x_2 = 1 \), \( y_2 = -1 \), \( z_2 = 1 \)

3. **Substitute into the Formula**: Substitute the coordinates into the distance formula:
\[ d = \sqrt{(1 - a)^2 + (-1 - 2)^2 + (1 - 1)^2} \]

4. **Simplify the Expression**: Calculate each component:
- \( (1 - a)^2 \)
- \( (-1 - 2)^2 = (-3)^2 = 9 \)
- \( (1 - 1)^2 = 0 \)

Therefore, the equation becomes:
\[ d = \sqrt{(1 - a)^2 + 9 + 0} = \sqrt{(1 - a)^2 + 9} \]

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

6. **Square Both Sides**: To eliminate the square root, square both sides of the equation:
\[ (1 - a)^2 + 9 = 25 \]

7. **Isolate the Squared Term**: Subtract 9 from both sides:
\[ (1 - a)^2 = 16 \]

8. **Take the Square Root**: Now take the square root of both sides:
\[ 1 - a = 4 \quad \text{or} \quad 1 - a = -4 \]

9. **Solve for \( a \)**:
- For \( 1 - a = 4 \):
\[ -a = 4 - 1 \implies -a = 3 \implies a = -3 \]
- For \( 1 - a = -4 \):
\[ -a = -4 - 1 \implies -a = -5 \implies a = 5 \]

10. **Final Values**: The possible values of \( a \) are:
\[ a = -3 \quad \text{and} \quad a = 5 \]

### Conclusion:
The values of \( a \) for which the distance between the points \( (a, 2, 1) \) and \( (1, -1, 1) \) is equal to 5 are \( a = -3 \) and \( a = 5 \).