Question

If the distance between the points (4, p) and (1, 0) is 5, then p is equal to :

• A. \( \pm 4 \)
• B. 4
• C. - 4
• D. 0

Hint:

Always remember to include both positive and negative results when solving for a squared variable unless a geometric constraint (like a distance or height) is mentioned for the variable itself.

Answer 1

The Correct Option is A

To solve this problem, we will use the distance formula to find the value of \( p \). The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Given points are \((4, p)\) and \((1, 0)\), and the distance between these points is \(5\). Substituting these into the formula gives:

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

Simplifying inside the square root:

\(5 = \sqrt{(-3)^2 + (-p)^2}\)

This can be simplified further to:

\(5 = \sqrt{9 + p^2}\)

To eliminate the square root, we square both sides of the equation:

\(25 = 9 + p^2\)

Subtract 9 from both sides to isolate the \( p^2 \) term:

\(p^2 = 16\)

Taking the square root of both sides gives:

\(p = \pm 4\)

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

The correct answer is \(\pm 4\), which matches the provided option.