Find the value of a if the distance between points `P(a,-8,4) and Q(-3,-5,4)` is 5.

Options

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

Answer

To find the value of \( a \) such that the distance between the points \( P(a, -8, 4) \) and \( Q(-3, -5, 4) \) is 5, we can use the distance formula for points in three-dimensional space.

### Step-by-Step Solution:

1. **Write 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:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

2. **Substitute the Coordinates**:
For points \( P(a, -8, 4) \) and \( Q(-3, -5, 4) \):
- \( x_1 = a \), \( y_1 = -8 \), \( z_1 = 4 \)
- \( x_2 = -3 \), \( y_2 = -5 \), \( z_2 = 4 \)

Substitute these values into the distance formula:
\[ d = \sqrt{((-3) - a)^2 + ((-5) - (-8))^2 + (4 - 4)^2} \]

3. **Simplify the Expression**:
Now simplify the expression:
\[ d = \sqrt{(-3 - a)^2 + (-5 + 8)^2 + (0)^2} \]
\[ d = \sqrt{(-3 - a)^2 + 3^2} \]
\[ d = \sqrt{(-3 - a)^2 + 9} \]

4. **Set the Distance Equal to 5**:
Since the distance is given as 5, we set up the equation:
\[ \sqrt{(-3 - a)^2 + 9} = 5 \]

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

6. **Isolate the Squared Term**:
Subtract 9 from both sides:
\[ (-3 - a)^2 = 16 \]

7. **Take the Square Root**:
Taking the square root of both sides gives us two equations:
\[ -3 - a = 4 \quad \text{or} \quad -3 - a = -4 \]

8. **Solve for \( a \)**:
- For the first equation:
\[ -3 - a = 4 \implies -a = 4 + 3 \implies -a = 7 \implies a = -7 \]
- For the second equation:
\[ -3 - a = -4 \implies -a = -4 + 3 \implies -a = -1 \implies a = -1 \]

9. **Final Values of \( a \)**:
The possible values of \( a \) are:
\[ a = -1 \quad \text{or} \quad a = -7 \]

### Conclusion:
Thus, the values of \( a \) that satisfy the condition are \( a = -1 \) and \( a = -7 \). The correct answer based on the context of the question is \( a = -7 \).