Distance between two point `(8, -4)` and `(0, a)` is `10`. All the values are in the same unit of length. Find the positive value of a.

Explanation

To find the positive value of \( a \) given the points \( (8, -4) \) and \( (0, a) \) with a distance of \( 10 \), we can follow these steps:

### Step 1: Use the Distance Formula
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, we have:
- Point 1: \( (x_1, y_1) = (8, -4) \)
- Point 2: \( (x_2, y_2) = (0, a) \)
- Given distance \( d = 10 \)

### Step 2: Substitute the Values into the Formula
Substituting the coordinates into the distance formula, we get:
\[ 10 = \sqrt{(0 - 8)^2 + (a - (-4))^2} \]
This simplifies to:
\[ 10 = \sqrt{(-8)^2 + (a + 4)^2} \]
\[ 10 = \sqrt{64 + (a + 4)^2} \]

### Step 3: Square Both Sides to Eliminate the Square Root
To eliminate the square root, we square both sides:
\[ 10^2 = 64 + (a + 4)^2 \]
\[ 100 = 64 + (a + 4)^2 \]

### Step 4: Isolate the Squared Term
Now, isolate \( (a + 4)^2 \):
\[ 100 - 64 = (a + 4)^2 \]
\[ 36 = (a + 4)^2 \]

### Step 5: Take the Square Root of Both Sides
Taking the square root of both sides gives:
\[ a + 4 = \pm 6 \]

### Step 6: Solve for \( a \)
Now, we have two cases to solve:
1. \( a + 4 = 6 \)
2. \( a + 4 = -6 \)

For the first case:
\[ a + 4 = 6 \implies a = 6 - 4 = 2 \]

For the second case:
\[ a + 4 = -6 \implies a = -6 - 4 = -10 \]

### Step 7: Choose the Positive Value
Since we are asked for the positive value of \( a \):
\[ \text{The positive value of } a = 2 \]

### Final Answer
The positive value of \( a \) is \( \boxed{2} \).

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