Values of a for which the distance between the
points (3, -5, 4) and (a, -8, 4) is 5 is

Options

A. `-1 or 7`
B. `2 or 7`
C. `3 or 1`
D. `2 or 1`

Answer

To find the values of \( a \) for which the distance between the points \( (3, -5, 4) \) and \( (a, -8, 4) \) is 5, we can use the distance formula in three-dimensional geometry.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( P = (3, -5, 4) \) and \( Q = (a, -8, 4) \).

2. **Use the Distance Formula**:
The distance \( d \) between two points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
In our case, we have:
- \( x_1 = 3 \), \( y_1 = -5 \), \( z_1 = 4 \)
- \( x_2 = a \), \( y_2 = -8 \), \( z_2 = 4 \)

3. **Set Up the Equation**:
We know the distance \( d = 5 \), so we can write:
\[ 5 = \sqrt{(a - 3)^2 + (-8 + 5)^2 + (4 - 4)^2} \]

4. **Simplify the Equation**:
Simplifying the terms inside the square root:
\[ 5 = \sqrt{(a - 3)^2 + (-3)^2 + 0^2} \]
This simplifies to:
\[ 5 = \sqrt{(a - 3)^2 + 9} \]

5. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ 25 = (a - 3)^2 + 9 \]

6. **Isolate the Squared Term**:
Rearranging gives:
\[ (a - 3)^2 = 25 - 9 \]
\[ (a - 3)^2 = 16 \]

7. **Take the Square Root**:
Taking the square root of both sides gives:
\[ a - 3 = \pm 4 \]

8. **Solve for \( a \)**:
This results in two equations:
- \( a - 3 = 4 \) which gives \( a = 7 \)
- \( a - 3 = -4 \) which gives \( a = -1 \)

9. **Final Values**:
Therefore, the values of \( a \) for which the distance is 5 are:
\[ a = 7 \quad \text{and} \quad a = -1 \]