Question

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

Hint:

Always remember to consider both the positive and negative square roots when solving equations of the form \( x^2 = k \).
In coordinate geometry, a point can lie either above or below the \( x \)-axis at the same horizontal distance, resulting in two valid symmetric solutions.

Answer 1

Step 1: Understanding the Question:
The problem asks us to find the value of the unknown coordinate \( p \) of a point \( (4, p) \), given that its distance from another point \( (1, 0) \) is exactly 5 units.
This is a standard application of coordinate geometry where we solve for an unknown variable using the distance condition.

Step 2: Key Formula or Approach:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a coordinate plane is given by the standard distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We can square both sides to eliminate the square root and obtain a quadratic equation in terms of \( p \):
\[ d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \]

Step 3: Detailed Explanation:
1. Let the two points be \( A(x_1, y_1) = (4, p) \) and \( B(x_2, y_2) = (1, 0) \).
2. The distance between them is given as \( d = 5 \).
3. Substitute these values into the squared distance formula:
\[ 5^2 = (1 - 4)^2 + (0 - p)^2 \]
4. Simplify the terms inside the parentheses:
\[ 25 = (-3)^2 + (-p)^2 \]
5. Calculate the squares:
\[ 25 = 9 + p^2 \]
6. Isolate the term \( p^2 \) by subtracting 9 from both sides of the equation:
\[ p^2 = 25 - 9 \]
\[ p^2 = 16 \]
7. Take the square root on both sides to find the values of \( p \):
\[ p = \pm \sqrt{16} \]
\[ p = 4 \quad \text{or} \quad p = -4 \]
8. Both positive and negative values are geometrically valid in a coordinate plane.

Step 4: Final Answer:
The values of \( p \) are \( 4 \) and \( -4 \) (or \( p = \pm 4 \)).