If the distance between the points (1,-8,a) and (-3,-5,4) is 5 units then find the value of 'a'.

Explanation

To find the value of 'a' given that the distance between the points (1, -8, a) and (-3, -5, 4) is 5 units, we can follow these steps:

### Step 1: Write down the distance formula for three-dimensional points.
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 the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 2: Substitute the given points into the formula.
Here, we have the points \((1, -8, a)\) and \((-3, -5, 4)\). We can substitute these values into the distance formula:
\[ d = \sqrt{((-3) - 1)^2 + ((-5) - (-8))^2 + (4 - a)^2} \]

### Step 3: Simplify the expression inside the square root.
Calculating the differences:
- For \(x\): \((-3) - 1 = -4\)
- For \(y\): \((-5) - (-8) = -5 + 8 = 3\)
- For \(z\): \(4 - a\)

Now, substituting these values back into the distance formula gives:
\[ d = \sqrt{(-4)^2 + (3)^2 + (4 - a)^2} \]

### Step 4: Square the distances.
Now we can square both sides of the equation since we know the distance \(d = 5\):
\[ 5^2 = (-4)^2 + (3)^2 + (4 - a)^2 \]
This simplifies to:
\[ 25 = 16 + 9 + (4 - a)^2 \]

### Step 5: Combine the constants.
Adding the constants on the right side:
\[ 25 = 25 + (4 - a)^2 \]

### Step 6: Isolate the square term.
Subtract 25 from both sides:
\[ 0 = (4 - a)^2 \]

### Step 7: Solve for 'a'.
Taking the square root of both sides gives:
\[ 4 - a = 0 \]
Thus,
\[ a = 4 \]

### Final Answer:
The value of \(a\) is \(4\).

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