In coordinate geometry, the distance between points (2, 7) and (k, −5) is 13 units. What is the correct value of k among the given options?

Introduction / Context:
This question applies the distance formula in the coordinate plane and then solves for an unknown x coordinate. It is typical of coordinate geometry problems in aptitude tests, where one point is partially unknown but the distance between two points is fixed.

Given Data / Assumptions:

• Point A has coordinates (2, 7).
• Point B has coordinates (k, −5).
• The distance between A and B is 13 units.
• We must determine the valid value of k from the answer options.

Concept / Approach:
The distance formula between points (x1, y1) and (x2, y2) is distance = √[(x2 − x1)^2 + (y2 − y1)^2]. Here the distance is known to be 13. We square both sides to remove the square root, producing an equation in k. Solving this quadratic equation gives possible values of k, after which we check which of these are present in the options.

Step-by-Step Solution:
Use the distance formula between (2, 7) and (k, −5). Distance^2 = (k − 2)^2 + (−5 − 7)^2. Compute the y difference: −5 − 7 = −12, so (−12)^2 = 144. Given distance = 13, so 13^2 = (k − 2)^2 + 144. Thus 169 = (k − 2)^2 + 144. Subtract 144 from both sides to get (k − 2)^2 = 25. Take square roots: k − 2 = ±5. So k = 2 + 5 = 7 or k = 2 − 5 = −3. Among the options, only k = 7 is listed.

Verification / Alternative check:
Check k = 7 by recomputing the distance. With B = (7, −5), the differences are 7 − 2 = 5 and −5 − 7 = −12. Then distance^2 = 5^2 + (−12)^2 = 25 + 144 = 169, so the distance is √169 = 13. Thus k = 7 is valid. Though k = −3 also satisfies the equation, it is not given among the choices, so it cannot be the answer for this multiple choice question.

Why Other Options Are Wrong:
The values −7, 6, −6, and 3 do not satisfy (k − 2)^2 = 25. For example, if k = 6, we get (6 − 2)^2 = 4^2 = 16, which leads to distance^2 = 16 + 144 = 160, not 169. Similar checks show that none of these choices yield a distance of 13 units from (2, 7).

Common Pitfalls:
A common error is to forget to square the full difference in the distance formula, or to confuse the roles of x and y coordinates. Another pitfall is ignoring the fact that squaring introduces two possible solutions, positive and negative roots. However, the multiple choice list may restrict which valid solution is acceptable as the final answer.

Final Answer:
The value of k that matches the given distance and appears in the options is 7.