Final answer:
The equation of the locus of points equidistant from (3,5) and (7,1) is the perpendicular bisector of the segment connecting the two points, which is y = x - 2.
Step-by-step explanation:
The question asks for the equation of the locus of points that are equidistant from the two points (3,5) and (7,1). This locus is a line called the perpendicular bisector of the segment connecting (3,5) and (7,1). To find the equation of the perpendicular bisector, we first find the midpoint of the segment by averaging the x-coordinates and the y-coordinates of the given points, which gives us the point (5,3). Next, we find the slope of the segment, which, with the points (3,5) and (7,1), is (1-5)/(7-3) = -1. The slope of the perpendicular bisector is the negative reciprocal of this slope, so it's 1. The equation of the perpendicular bisector can now be found using the point-slope form with the midpoint and the slope of the bisector:
y - y1 = m(x - x1)
Substituting the midpoint (5,3) and the slope 1, we have:
y - 3 = 1(x - 5)
This simplifies to:
y = x - 2
The equation of the locus of points equidistant from (3,5) and (7,1) is y = x - 2.