To find the coordinates of the foot of the normal to the parabola y^2 = 3x, which is perpendicular to the line y = 2x + 4, we can use the properties of perpendicular lines and the concept of the slope of a line.

Let's break down the problem into steps:

Step 1: Find the slope of the given line.
The given line is y = 2x + 4, which is in the slope-intercept form y = mx + b, where m is the slope of the line. Comparing this equation with the standard form, we can see that the slope of the line is 2.

Step 2: Find the slope of the normal to the parabola.
The slope of the normal to the parabola is the negative reciprocal of the slope of the tangent to the parabola at a given point. To find this slope, we need to find the tangent to the parabola at a point.

Step 3: Find the point of tangency.
To find the point of tangency, we can equate the slope of the tangent to the slope of the parabola at a given point. Let's consider a point (h, k) on the parabola. The slope of the parabola at this point can be found by differentiating the equation of the parabola with respect to x.

Differentiating y^2 = 3x with respect to x, we get:
2yy' = 3
y' = 3/(2y)

Since the slope of the tangent is the derivative of the y-coordinate with respect to the x-coordinate, we can substitute the coordinates of the point (h, k) into y' to find the slope of the tangent at that point.

Substituting h and k into y', we get:
m_tangent = 3/(2k)

Step 4: Find the slope of the normal.
The slope of the normal is the negative reciprocal of the slope of the tangent. Therefore, the slope of the normal is:
m_normal = -2k/3

Step 5: Find the point of intersection.
We know that the normal to the parabola is perpendicular to the given line y = 2x + 4. Therefore, the product of the slopes of the two lines should be -1.

(-2k/3) * 2 = -1
-4k/3 = -1
k = 3/4

Substituting k = 3/4 into the equation of the parabola, we can solve for h:

(3/4)^2 = 3h
9/16 = 3h
h = 3/16

Therefore, the point of intersection is (3/16, 3/4).

Step 6: Find the foot of the normal.
The foot of the normal is the point on the parabola that is closest to the given line. Since the normal is perpendicular to the line, the foot of the normal is the projection of the point of intersection onto the line.

To find the foot of the normal, we need to find the equation of the line passing through the point of intersection (3/16, 3/4) and perpendicular to the given line y = 2x + 4.

The equation of a line passing through a point (x1, y