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.
$x=$
We are given two points on a Cartesian plane: \(A = (16, 11)\) and \(B = (x, -4)\). We are told that the distance between these two points is \(17\). The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Our goal is to find all possible values of \(x\) that satisfy this condition.
Substituting the coordinates of points A and B and the distance \(d = 17\) into the distance formula, we get:
\[17 = \sqrt{(x - 16)^2 + (-4 - 11)^2}\]
To simplify this equation, we first square both sides to eliminate the square root:
\[17^2 = (x - 16)^2 + (-4 - 11)^2\]
\[289 = (x - 16)^2 + (-15)^2\]
Since \((-15)^2 = 225\), the equation becomes:
\[289 = (x - 16)^2 + 225\]
Next, we isolate the term \((x - 16)^2\) by subtracting 225 from both sides:
\[(x - 16)^2 = 289 - 225\]
\[(x - 16)^2 = 64\]
Now, we take the square root of both sides to solve for \(x - 16\). Remember that taking the square root of a square results in two possible cases (positive and negative):
\[x - 16 = 8 \quad \text{or} \quad x - 16 = -8\]
Now we solve the two resulting linear equations:
Thus, the possible values for \(x\) are \(8\) and \(24\). The final answer is \(\boxed{8, 24}\).
The distance formula is a fundamental tool in coordinate geometry, used in various fields like navigation, robotics, and computer graphics. For example, if you are tracking a drone's position, the distance between its current location and its base can be calculated using this formula. Understanding how to solve for an unknown coordinate given a distance allows you to determine potential locations of an object when you only know its distance from a reference point, which is essential for tasks like GPS triangulation or motion planning.