Explanation:
To find the locus of the midpoints of the perpendiculars drawn from points on the line x = 2y to the line x = y, we can start by considering an arbitrary point (x, y) on the line x = 2y.

1. Finding the equation of the line perpendicular to x = y:
Since the line x = y has a slope of 1, the line perpendicular to it will have a slope of -1 (negative reciprocal).

Using the point-slope form of a line equation, the equation of the line perpendicular to x = y passing through the point (x, y) is given by:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is the point on the line.

In this case, the equation of the line perpendicular to x = y passing through (x, y) is:
y - y = -1(x - 2y)
Simplifying, we get:
y + x = 2y
x - y = 0

2. Finding the midpoint of the line segment:
To find the midpoint of the line segment between the point (x, y) on x = 2y and the intersection point of the perpendicular line and x = y, we can solve the system of equations formed by the two lines.

The equations are:
x = 2y (given line)
x - y = 0 (perpendicular line)

Solving these equations, we get the point of intersection as (2, 1).

The midpoint of the line segment between (x, y) and (2, 1) is given by:
[(x + 2)/2, (y + 1)/2]

3. Finding the equation of the locus:
To find the locus of the midpoints, we substitute x = 2y into the expression for the midpoint coordinates:
[(2y + 2)/2, (y + 1)/2]
Simplifying, we get:
[y + 1, (y + 1)/2]
2(y + 1) = y + 1

Expanding and simplifying, we get:
2y + 2 = y + 1
y = -1

Substituting y = -1 into x = 2y, we get:
x = 2(-1)
x = -2

Therefore, the locus of the midpoints is the point (-2, -1), which can be written in the form of a line equation as:
x + 2y = 0

Comparing this equation with the given options, we can see that the correct answer is option 'A': 5x - 7y = 0.