If y is a positive integer such that the distance between the points (-6, -1) and (-6, y) is 12 units, then y= 

Options

A. 5
B. 8
C. 11
D. 1

Answer

To solve the problem step by step, we will find the value of \( y \) given that the distance between the points \((-6, -1)\) and \((-6, y)\) is 12 units.

### Step 1: Understand 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: Identify the coordinates
In our case, the coordinates of the points are:
- Point A: \((-6, -1)\) (where \( x_1 = -6 \) and \( y_1 = -1 \))
- Point B: \((-6, y)\) (where \( x_2 = -6 \) and \( y_2 = y \))

### Step 3: Substitute into the distance formula
Substituting the coordinates into the distance formula:
\[ d = \sqrt{((-6) - (-6))^2 + (y - (-1))^2} \]
This simplifies to:
\[ d = \sqrt{(0)^2 + (y + 1)^2} \]
Thus, we have:
\[ d = \sqrt{(y + 1)^2} \]

### Step 4: Set the distance equal to 12
According to the problem, the distance is 12 units:
\[ \sqrt{(y + 1)^2} = 12 \]

### Step 5: Remove the square root
Since the square root of a square gives the absolute value, we can write:
\[ |y + 1| = 12 \]

### Step 6: Solve the absolute value equation
This gives us two cases to consider:
1. \( y + 1 = 12 \)
2. \( y + 1 = -12 \)

#### Case 1:
\[ y + 1 = 12 \implies y = 12 - 1 = 11 \]

#### Case 2:
\[ y + 1 = -12 \implies y = -12 - 1 = -13 \]

### Step 7: Determine the valid solution
Since the problem states that \( y \) is a positive integer, we discard \( y = -13 \) and keep:
\[ y = 11 \]

### Final Answer
Thus, the value of \( y \) is:
\[ \boxed{11} \]