The distance between the points (4,8) and (k, - 4) is 13 . What is the value of k ?

Options

A. 1
B. 3
C. `-1`
D. `-3`

Answer

To find the value of \( k \) such that the distance between the points \( (4, 8) \) and \( (k, -4) \) is 13, we can 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 points
Let:
- \( (x_1, y_1) = (4, 8) \)
- \( (x_2, y_2) = (k, -4) \)

### Step 2: Substitute into the distance formula
We know the distance is 13, so we can set up the equation:

\[ 13 = \sqrt{(k - 4)^2 + (-4 - 8)^2} \]

### Step 3: Simplify the equation
Calculate \( -4 - 8 \):

\[ -4 - 8 = -12 \]

Now, substitute this back into the equation:

\[ 13 = \sqrt{(k - 4)^2 + (-12)^2} \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:

\[ 13^2 = (k - 4)^2 + 144 \]

Calculating \( 13^2 \):

\[ 169 = (k - 4)^2 + 144 \]

### Step 5: Isolate the squared term
Subtract 144 from both sides:

\[ 169 - 144 = (k - 4)^2 \]

This simplifies to:

\[ 25 = (k - 4)^2 \]

### Step 6: Take the square root of both sides
Now, we take the square root of both sides:

\[ k - 4 = \pm 5 \]

### Step 7: Solve for \( k \)
This gives us two equations to solve:

1. \( k - 4 = 5 \)
2. \( k - 4 = -5 \)

For the first equation:

\[ k = 5 + 4 = 9 \]

For the second equation:

\[ k = -5 + 4 = -1 \]

### Final Answer
Thus, the possible values for \( k \) are \( k = 9 \) and \( k = -1 \).

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