Final answer:
The line that has the maximum distance from the point (3, 1) will be perpendicular to the line segment joining (1, 2) and (3, 1). The precise equation is not among the options, but the line with the closest, steepest slope is option d, y = -2x + 5.
Step-by-step explanation:
The question asks for the equation of a line that goes through the point (1, 2) and is at the maximum distance from the point (3, 1). To find this, we need to consider the perpendicular distance from the point (3, 1) to the line passing through (1, 2). The line with the greatest perpendicular distance will be the one that is perpendicular to the line segment connecting the two points.
The slope of the line segment connecting (1, 2) and (3, 1) is (1-2)/(3-1) = -1/2. Therefore, the slope of the line perpendicular to this would be the negative reciprocal, which is 2. Now let's find the equation of the line with slope 2 that passes through (1, 2): y - 2 = 2(x - 1), which simplifies to y = 2x - 2 + 2, so y = 2x. But this equation is not among the options given. Instead, we look for the one with the closest slope to 2, since none of the provided options exactly match the perpendicular slope. The closest option is option d, y = -2x + 5, which, while not exactly perpendicular, has the steepest slope and would likely result in the maximum distance compared to the other options provided.