In coordinate geometry, the distance between points (7, 7) and (k, -5) is 13 units. Using the distance formula, determine which option gives a possible value of k.

Introduction / Context:
This question checks your understanding of the distance formula in the Cartesian plane. Given one point with fixed coordinates and another point with an unknown x-coordinate k, you are told the distance between them and asked to find which value of k from the options makes that distance equal to 13 units. This is a classic coordinate geometry and aptitude-style problem involving simple quadratic reasoning.

Given Data / Assumptions:

• First point: (7, 7).
• Second point: (k, −5).
• The distance between the points is 13 units.
• We must find a value of k from the options that satisfies this condition.

Concept / Approach:
We apply the distance formula between two points (x1, y1) and (x2, y2): Distance^2 = (x2 − x1)^2 + (y2 − y1)^2. Here, the distance is 13, so its square is 169. We substitute x1 = 7, y1 = 7, x2 = k, and y2 = −5 into the formula, simplify to get an equation in k, and then solve for k. After finding the possible values, we see which one appears in the options given and choose that as the answer.

Step-by-Step Solution:
1) Use the distance formula: (k − 7)^2 + (−5 − 7)^2 = 13^2. 2) Simplify the y-difference: −5 − 7 = −12, so the equation becomes (k − 7)^2 + (−12)^2 = 169. 3) Compute (−12)^2 = 144, giving (k − 7)^2 + 144 = 169. 4) Subtract 144 from both sides: (k − 7)^2 = 169 − 144 = 25. 5) Take square roots: k − 7 = ±5, so k = 7 + 5 = 12 or k = 7 − 5 = 2. 6) The possible k values from the equation are 2 and 12. Among the given options −4, −2, 2, 4, and 11, the only matching value is k = 2.

Verification / Alternative check:
Verify k = 2 by computing the distance between (7, 7) and (2, −5). The difference in x is 2 − 7 = −5, and the difference in y is −5 − 7 = −12. The distance squared is (−5)^2 + (−12)^2 = 25 + 144 = 169, so the distance is √169 = 13 units, which matches the requirement exactly. This confirms that k = 2 is indeed a valid solution consistent with the problem statement.

Why Other Options Are Wrong:
If k = −4, −2, 4, or 11, substituting into the distance formula will not produce a distance of 13 units. For example, if k = 4, then (4 − 7)^2 + (−12)^2 = (−3)^2 + 144 = 9 + 144 = 153, which gives a distance of √153, not 13. Similarly, the other values do not satisfy the equation (k − 7)^2 = 25. Therefore, they do not correspond to a distance of 13 units between the two points.

Common Pitfalls:
A common error is forgetting to square the differences or forgetting that the distance formula uses the square of the distance, not the distance itself, when setting up the equation. Another mistake is to take only the positive square root when solving (k − 7)^2 = 25, thereby missing one of the possible values. Careful application of the formula, along with proper handling of squares and square roots, ensures accurate results.

Final Answer:
The possible value of k among the options that makes the distance exactly 13 units is 2.