Find the vaue of x if the distance between the points (2, -11) and (x, -3) is 10 unit.

Answer

To find the value of \( x \) such that the distance between the points \( (2, -11) \) and \( (x, -3) \) is 10 units, we will use the distance formula. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 1: Identify the coordinates
Let the points be:
- \( (x_1, y_1) = (2, -11) \)
- \( (x_2, y_2) = (x, -3) \)

### Step 2: Substitute the coordinates into the distance formula
The distance between the points is given to be 10 units. Therefore, we can set up the equation:

\[ 10 = \sqrt{(x - 2)^2 + (-3 - (-11))^2} \]

### Step 3: Simplify the equation
First, simplify the expression inside the square root:

\[ -3 - (-11) = -3 + 11 = 8 \]

Now, substituting back into the equation gives:

\[ 10 = \sqrt{(x - 2)^2 + 8^2} \]

This simplifies to:

\[ 10 = \sqrt{(x - 2)^2 + 64} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides results in:

\[ 10^2 = (x - 2)^2 + 64 \]

This simplifies to:

\[ 100 = (x - 2)^2 + 64 \]

### Step 5: Isolate the squared term
Subtract 64 from both sides:

\[ 100 - 64 = (x - 2)^2 \]

This simplifies to:

\[ 36 = (x - 2)^2 \]

### Step 6: Take the square root of both sides
Taking the square root gives:

\[ \sqrt{36} = |x - 2| \]

This results in two equations:

\[ x - 2 = 6 \quad \text{or} \quad x - 2 = -6 \]

### Step 7: Solve for \( x \)
1. For \( x - 2 = 6 \):
\[ x = 6 + 2 = 8 \]

2. For \( x - 2 = -6 \):
\[ x = -6 + 2 = -4 \]

### Conclusion
Thus, the values of \( x \) that satisfy the condition are:

\[ x = 8 \quad \text{or} \quad x = -4 \]