If the distance between the points (a,0,1) and (0,1,2) is `sqrt27` then the value of a is

Options

A. 5
B. `+-5`
C. `-5`
D. none of these

Explanation

To solve the problem, we need to find the value of \( a \) given the distance between the points \( (a, 0, 1) \) and \( (0, 1, 2) \) is \( \sqrt{27} \).

### Step-by-Step Solution:

1. **Identify the Points**:
We have two points:
- Point 1: \( P_1 = (a, 0, 1) \)
- Point 2: \( P_2 = (0, 1, 2) \)

2. **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:
\[ d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2} \]
For our points, this becomes:
\[ d = \sqrt{(a - 0)^2 + (0 - 1)^2 + (1 - 2)^2} \]

3. **Substituting the Known Distance**:
We know that the distance \( d \) is \( \sqrt{27} \):
\[ \sqrt{27} = \sqrt{a^2 + (0 - 1)^2 + (1 - 2)^2} \]

4. **Simplifying the Equation**:
Now, simplify the equation:
\[ \sqrt{27} = \sqrt{a^2 + (-1)^2 + (-1)^2} \]
\[ \sqrt{27} = \sqrt{a^2 + 1 + 1} \]
\[ \sqrt{27} = \sqrt{a^2 + 2} \]

5. **Squaring Both Sides**:
To eliminate the square root, square both sides:
\[ 27 = a^2 + 2 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ a^2 = 27 - 2 \]
\[ a^2 = 25 \]

7. **Finding the Values of \( a \)**:
Taking the square root of both sides:
\[ a = \pm \sqrt{25} \]
\[ a = \pm 5 \]

8. **Conclusion**:
Therefore, the possible values of \( a \) are:
\[ a = 5 \quad \text{or} \quad a = -5 \]

### Final Answer:
The value of \( a \) is \( 5 \) or \( -5 \).

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