Final answer:
The solution involves finding points at a certain distance from a line and a point, which leads to intersections of a circle and lines parallel to y-axis. The symmetry around the line y=13 and the vertical line through (7,13) simplifies the sum calculation to 80. To get specific coordinates, a calculation involving the equations of a circle and the distance from a line is necessary.
Step-by-step explanation:
The given problem involves finding points that are equidistant from a line and a specific point. The line in question is y=13, so any point 5 units away from this line would be on one of the lines y=8 or y=18, considering the distance above and below the line. The point given is (7,13), and the distance required from this point is 13 units, which forms a circle centered at this point with a radius of 13. The points of interest are the intersections of the circle with the two lines mentioned. We can solve for these intersections using the distance formula for a point to a line, and the standard equation for a circle. The key here is that the points will be symmetrical about the line y=13 and the vertical line through the point (7,13). Since the question asks for the sum of the x- and y-coordinates, the symmetry allows us to state that each opposite pair of points will have the same y-coordinate and x-coordinates that are equidistant from 7. Therefore, while the individual coordinates depend on exact values from the intersections, their sum will be four times the y-coordinate of the line y=13, which is 52, plus four times the x-coordinate of the point (7,13), which is 28, leading to a total sum of 80. However, to provide a specific numerical solution we would need to calculate the exact points of intersection which requires setting up and solving the respective equations for a circle and a line.