If the distance between the points (4,p) and (1,0) is 5 , then find the value of p.

Options

A. 4 only
B. `+-4`
C. `- 4 only`
D. 0

Explanation

To find the value of \( p \) given that the distance between the points \( (4, p) \) 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
Let:
- \( (x_1, y_1) = (4, p) \)
- \( (x_2, y_2) = (1, 0) \)

### Step 2: Substitute the coordinates into the distance formula
We know that the distance \( d \) is 5, so we set up the equation:

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

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

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

Now substituting this back into the equation:

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

This simplifies to:

\[ 5 = \sqrt{9 + p^2} \]

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

\[ 25 = 9 + p^2 \]

### Step 5: Solve for \( p^2 \)
Subtract 9 from both sides:

\[ 25 - 9 = p^2 \]
\[ 16 = p^2 \]

### Step 6: Find the value of \( p \)
Taking the square root of both sides, we get:

\[ p = \pm 4 \]

Thus, the possible values of \( p \) are \( 4 \) and \( -4 \).

### Final Answer:
The values of \( p \) are \( 4 \) and \( -4 \).

---