Final answer:
The relation between x and y for point P (x, y) that lies on the perpendicular bisector of the line joining (7, 1) and (3, 5) is given by the equation y = x - 2.
Step-by-step explanation:
To find the relation between x and y if point P (x, y) lies on the perpendicular bisector of the line joining the points (7, 1) and (3, 5), we first need to find the midpoint and the slope of the line segment joining these two points.
The midpoint M can be calculated as M = ((7 + 3)/2, (1 + 5)/2) = (5, 3).
The slope of the line through points (7, 1) and (3, 5) is (5 - 1)/(3 - 7) = 4/(-4) = -1.
The slope of the perpendicular bisector will be the negative reciprocal of -1, which is 1.
Now, using the point-slope form of the equation of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Plugging in our values, we get y - 3 = 1(x - 5), which simplifies to y = x - 2.
Therefore, the equation of the perpendicular bisector that the point P (x, y) lies on is y = x - 2.