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:

The coordinates form a right-angled triangle where the hypotenuse is the distance (5) and the horizontal side is the \(x\)-difference (\(|4 - 1| = 3\)).
Using the standard \((3, 4, 5)\) Pythagorean triple, the vertical side (which is the \(y\)-difference \(|p - 0|\)) must be equal to 4:
\[ |p| = 4 \implies p = \pm 4 \]
This geometric approach is extremely quick and visual!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given two points in a Cartesian plane: \((4, p)\) and \((1, 0)\). The straight-line distance between these points is given as 5 units. We need to determine the value(s) of the unknown \(y\)-coordinate \(p\).

Step 2: Key Formula or Approach:
The distance \(d\) between two coordinate points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the standard distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Detailed Explanation:
1. Let the two points be:
\[ (x_1, y_1) = (4, p) \]
\[ (x_2, y_2) = (1, 0) \]
The distance \(d = 5\).
2. Substitute these values into the distance formula:
\[ 5 = \sqrt{(1 - 4)^2 + (0 - p)^2} \]
3. Simplify the terms inside the square root:
- The \(x\)-difference squared: \((1 - 4)^2 = (-3)^2 = 9\)
- The \(y\)-difference squared: \((0 - p)^2 = (-p)^2 = p^2\)
Substitute these:
\[ 5 = \sqrt{9 + p^2} \]
4. Square both sides of the equation to clear the square root:
\[ 5^2 = 9 + p^2 \]
\[ 25 = 9 + p^2 \]
5. Isolate \(p^2\):
\[ p^2 = 25 - 9 \]
\[ p^2 = 16 \]
6. Take the square root of both sides to find \(p\):
\[ p = \pm \sqrt{16} \]
\[ p = \pm 4 \]
Therefore, \(p\) can be either \(+4\) or \(-4\).

Step 4: Final Answer:
The value of \(p\) is \(\pm 4\), which corresponds to option (A).