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

Options

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

Explanation

To find the value of \( p \) such that the distance between the points \( (4, p) \) 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 \( (4, p) \) and \( (1, 0) \).

### Step 2: Substitute the Coordinates
Substituting the coordinates into the distance formula:
\[ d = \sqrt{(1 - 4)^2 + (0 - p)^2} \]
This simplifies to:
\[ d = \sqrt{(-3)^2 + (-p)^2} \]
\[ d = \sqrt{9 + p^2} \]

### Step 3: Set the Distance Equal to 5
According to the problem, the distance is 5. Therefore, we set up the equation:
\[ \sqrt{9 + p^2} = 5 \]

### Step 4: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ 9 + p^2 = 25 \]

### Step 5: Solve for \( p^2 \)
Now, we can isolate \( p^2 \):
\[ p^2 = 25 - 9 \]
\[ p^2 = 16 \]

### Step 6: Find \( p \)
Taking the square root of both sides gives us:
\[ p = \pm 4 \]

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

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