Given:
- A straight line parallel to y = √3x
- The line passes through Q(2,3)
- The line intersects the line 2x + 4y - 27 = 0 at point P

To Find:
- The length of PQ

Steps to Solve:
1. Determine the equation of the line parallel to y = √3x that passes through point Q(2,3).
2. Find the coordinates of point P by solving the system of equations formed by the parallel line and the given line 2x + 4y - 27 = 0.
3. Calculate the distance between points P and Q to find the length of PQ.

Solution:

Step 1: Determine the equation of the parallel line
The given line y = √3x can be rewritten in slope-intercept form as y = mx + b, where m is the slope and b is the y-intercept. Comparing the given equation with the slope-intercept form, we can determine that the slope of the line is √3.

Since the parallel line has the same slope (√3) as the given line, the equation of the parallel line passing through point Q(2,3) can be written as:
y - 3 = √3(x - 2)

Step 2: Find the coordinates of point P
To find the coordinates of point P, we need to solve the system of equations formed by the parallel line and the given line 2x + 4y - 27 = 0.

Substituting y from the equation of the parallel line into the given line equation, we get:
2x + 4(√3(x - 2)) - 27 = 0

Simplifying the equation:
2x + 4√3x - 8√3 - 27 = 0
(2 + 4√3)x = 8√3 + 27
x = (8√3 + 27)/(2 + 4√3)

Substituting the value of x back into the equation of the parallel line, we can solve for y:
y - 3 = √3((8√3 + 27)/(2 + 4√3) - 2)

Simplifying the equation:
y - 3 = √3(8√3 - 4√3 + 27 - 4(2))/(2 + 4√3)
y - 3 = √3(8 - 4 + 27 - 8)/(2 + 4√3)
y - 3 = √3(23)/(2 + 4√3)
y = 3 + √3(23)/(2 + 4√3)

Therefore, the coordinates of point P are:
P(x, y) = ( (8√3 + 27)/(2 + 4√3), 3 + √3(23)/(2 + 4√3) )

Step 3: Calculate the length of PQ
The distance between two points P(x1, y1) and Q(x2, y2) can be calculated using the distance formula