If the distance between the points (4, k) and (1, 0) is 5, what can be the possible values of k ?

Answer

To find the possible values of \( k \) such that the distance between the points \( (4, k) \) and \( (1, 0) \) is 5, 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
Here, the points are \( (x_1, y_1) = (4, k) \) and \( (x_2, y_2) = (1, 0) \).

### Step 2: Set up the distance equation
According to the problem, the distance between these points is 5. Therefore, we can set up the equation:

\[ \sqrt{(1 - 4)^2 + (0 - k)^2} = 5 \]

### Step 3: Simplify the equation
Calculating \( (1 - 4)^2 \):

\[ (1 - 4)^2 = (-3)^2 = 9 \]

Now, substituting this back into the equation gives:

\[ \sqrt{9 + (0 - k)^2} = 5 \]

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

\[ 9 + (0 - k)^2 = 5^2 \]

This simplifies to:

\[ 9 + k^2 = 25 \]

### Step 5: Solve for \( k^2 \)
Now, we can isolate \( k^2 \):

\[ k^2 = 25 - 9 \]

\[ k^2 = 16 \]

### Step 6: Find the values of \( k \)
Taking the square root of both sides gives us:

\[ k = \pm 4 \]

Thus, the possible values of \( k \) are:

\[ k = 4 \quad \text{or} \quad k = -4 \]

### Final Answer
The possible values of \( k \) are \( 4 \) and \( -4 \).

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