The distance between the points with coordinates (4, 8) and (k, -4) is 13 units. What is the possible value of k?

Introduction / Context:

This question checks understanding of the distance formula in coordinate geometry and the ability to work with algebraic expressions. We are given two points in the coordinate plane, one of which has an unknown x coordinate k, and we are told that the distance between these points is 13 units. Using the distance formula and the given coordinates, we must find the correct value of k that satisfies this geometric condition. This type of problem commonly appears in school mathematics and aptitude tests to reinforce algebra and geometry together.

Given Data / Assumptions:

• First point A has coordinates (4, 8).
• Second point B has coordinates (k, -4).
• The distance between A and B is 13 units.
• We assume a standard Cartesian coordinate system.
• We need the value of k that satisfies the distance formula.

Concept / Approach:

In a two dimensional Cartesian plane, the distance between points (x1, y1) and (x2, y2) is given by the formula distance^2 = (x2 - x1)^2 + (y2 - y1)^2. If the distance is known, we can square it and set up an equation: (x2 - x1)^2 + (y2 - y1)^2 = given distance squared. In this problem we substitute x1 = 4, y1 = 8, x2 = k and y2 = -4, then equate the expression to 13^2. Solving the resulting quadratic equation in k gives possible values for k. Finally, we choose the value that appears in the options.

Step-by-Step Solution:

Verification / Alternative check:

We now check which of these k values appear in the options. The options include -1 but do not include 9. For k = -1, the points become (4, 8) and (-1, -4). The horizontal difference is 4 - (-1) = 5 units and the vertical difference is 8 - (-4) = 12 units. The distance squared is 5^2 + 12^2 = 25 + 144 = 169, so the distance is 13 units. This confirms that k = -1 satisfies the condition and is compatible with the given choices.

Why Other Options Are Wrong:

If k = 1, then (k - 4)^2 = (-3)^2 = 9, and the total distance squared would be 9 + 144 = 153, giving a distance less than 13. For k = 3, we would get (3 - 4)^2 = 1, and the distance squared would become 1 + 144 = 145, still not 169. For k = -3, we get (-3 - 4)^2 = (-7)^2 = 49, and the distance squared would be 49 + 144 = 193, which is greater than 169. Therefore none of these values produce 13 as the distance.

Common Pitfalls:

Common mistakes include forgetting to square the differences correctly, especially the vertical difference, or forgetting that we must square the distance 13 to obtain 169 in the equation. Some learners also mistake 13^2 for 13 * 2 or incorrectly handle negative signs when computing differences like -4 - 8. Another pitfall is not checking which roots of the quadratic equation are actually present in the options and accidentally using a value not listed.

Final Answer:

The valid value of k from the given options is -1.