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

Answer

To find the possible values of \( k \) given that the distance between the points \( (4, k) \) and \( (1, 0) \) is 5, we can follow these steps:

### Step 1: Use the Distance Formula
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, the points are \( (4, k) \) and \( (1, 0) \), and the distance is given as 5.

### Step 2: Substitute the Points into the Formula
Substituting the coordinates into the distance formula, we have:
\[ 5 = \sqrt{(1 - 4)^2 + (0 - k)^2} \]

### Step 3: Simplify the Equation
Calculating \( (1 - 4)^2 \):
\[ 1 - 4 = -3 \quad \Rightarrow \quad (-3)^2 = 9 \]
Thus, the equation becomes:
\[ 5 = \sqrt{9 + (0 - k)^2} \]
This simplifies to:
\[ 5 = \sqrt{9 + k^2} \]

### Step 4: Square Both Sides
To eliminate the square root, we square both sides:
\[ 5^2 = 9 + k^2 \]
This gives us:
\[ 25 = 9 + k^2 \]

### Step 5: Solve for \( k^2 \)
Now, we isolate \( k^2 \):
\[ k^2 = 25 - 9 \]
Calculating the right side:
\[ k^2 = 16 \]

### Step 6: Find \( k \)
To find \( k \), we take the square root of both sides:
\[ k = \pm 4 \]
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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