Given $(16,11)$ and $(x,-4)$, find all $x$ such that the distance between these two points is 17. Separate multiple answers with a comma.
We are given two points in the Cartesian plane: \(P_1 = (16, 11)\) and \(P_2 = (x, -4)\). We are also given that the distance between these two points is \(d = 17\). The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is defined as:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Our goal is to solve for all possible values of \(x\) that satisfy this condition.
Substitute the coordinates of the given points and the distance \(d=17\) into the distance formula:
\[17 = \sqrt{(x - 16)^2 + (-4 - 11)^2}\]
Simplifying the expression inside the square root:
\[17 = \sqrt{(x - 16)^2 + (-15)^2}\]
\[17 = \sqrt{(x - 16)^2 + 225}\]
Now, square both sides to remove the square root:
\[17^2 = (x - 16)^2 + 225\]
\[289 = (x - 16)^2 + 225\]
To solve for \(x\), first isolate the term containing \(x\):
\[(x - 16)^2 = 289 - 225\]
\[(x - 16)^2 = 64\]
Taking the square root of both sides gives us two possibilities:
\[x - 16 = 8 \quad \text{or} \quad x - 16 = -8\]
Solving each case:
Case 1: \(x - 16 = 8 \implies x = 16 + 8 = 24\)
Case 2: \(x - 16 = -8 \implies x = 16 - 8 = 8\)
Thus, the possible values for \(x\) are \(24\) and \(8\).
Understanding distance in the Cartesian plane is fundamental to coordinate geometry. For example, if you are tracking the flight path of two drones on a grid where each unit represents one kilometer, and you know their coordinates, you can calculate the distance between them. If one drone is at (16, 11) and the second drone is at (x, -4) and you know they are 17 kilometers apart, you can determine the possible locations for the second drone using the distance formula. This is a crucial skill for navigation, robotics, and mapping applications.