Find the value of 'a', if the distance between the points A(-3,14) and B (a,5) is 9 units.

Answer

To find the value of 'a' such that the distance between the points A(-3, 14) and B(a, 5) is 9 units, we can follow these steps:

### Step 1: Write down the distance formula.
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 2: Assign the coordinates.
Let \( A(-3, 14) \) be \( (x_1, y_1) \) and \( B(a, 5) \) be \( (x_2, y_2) \). Thus, we have:
- \( x_1 = -3 \)
- \( y_1 = 14 \)
- \( x_2 = a \)
- \( y_2 = 5 \)

### Step 3: Substitute the coordinates into the distance formula.
We know the distance \( d \) is 9 units, so we set up the equation:
\[ 9 = \sqrt{(a - (-3))^2 + (5 - 14)^2} \]
This simplifies to:
\[ 9 = \sqrt{(a + 3)^2 + (-9)^2} \]

### Step 4: Simplify the equation.
Now we can simplify the equation:
\[ 9 = \sqrt{(a + 3)^2 + 81} \]

### Step 5: Square both sides to eliminate the square root.
Squaring both sides gives:
\[ 9^2 = (a + 3)^2 + 81 \]
\[ 81 = (a + 3)^2 + 81 \]

### Step 6: Subtract 81 from both sides.
\[ 81 - 81 = (a + 3)^2 \]
\[ 0 = (a + 3)^2 \]

### Step 7: Solve for 'a'.
Taking the square root of both sides gives:
\[ a + 3 = 0 \]
Thus,
\[ a = -3 \]

### Conclusion
The value of \( a \) is \( -3 \).