Given Information:
- A straight line makes an angle of 30 degrees with the x-axis.
- The line passes through a point P(3,4).
- The line intersects the line 12x + 5y + 10 = 0 at point Q.

To Find:
The length of PQ.

Approach:
1. Determine the equation of the straight line passing through point P and making an angle of 30 degrees with the x-axis.
2. Find the coordinates of point Q by solving the system of equations formed by the given line and the equation of the straight line determined in step 1.
3. Calculate the distance between points P and Q using the distance formula.

Solution:

Step 1: Determine the Equation of the Straight Line
Let the equation of the straight line passing through point P be y = mx + c, where m is the slope and c is the y-intercept.

Given that the line makes an angle of 30 degrees with the x-axis, we can use the tangent of the angle to find the slope.

The tangent of 30 degrees is √3/3, which is equal to the slope (m) of the line.

So, we have the equation:
y = (√3/3)x + c

To find the value of c, substitute the coordinates of point P into the equation:
4 = (√3/3)(3) + c
4 = √3 + c
c = 4 - √3

Therefore, the equation of the straight line passing through point P is:
y = (√3/3)x + (4 - √3)

Step 2: Find the Coordinates of Point Q
Substitute the equation of the straight line into the equation of the given line to find the coordinates of point Q.

12x + 5y + 10 = 0
12x + 5((√3/3)x + (4 - √3)) + 10 = 0
12x + (√3/3)x + 5(4 - √3) + 10 = 0
(12 + (√3/3))x + 30 - 5√3 + 10 = 0
(12 + (√3/3))x - 5√3 + 40 = 0

Solve the equation to find the value of x:
(12 + (√3/3))x = 5√3 - 40
x = (5√3 - 40)/(12 + (√3/3))

Substitute the value of x into the equation of the straight line to find the value of y:
y = (√3/3)x + (4 - √3)
y = (√3/3)((5√3 - 40)/(12 + (√3/3))) + (4 - √3)

Simplify the equation to find the value of y.

Step 3: Calculate the Distance between Points P and Q
We now have the coordinates of points P and Q. Use the distance formula to find the length of PQ:

Distance PQ = √((x2 - x1)^2 + (y2 - y1)^2)

Substitute the