A line forms a triangle of area `54sqrt(3)s q u a r e` units with the coordinate axes. Find the equation of the line if the perpendicular drawn from the origin to the line makes an angle of `60^0` with the X-axis.

Explanation

To find the equation of the line that forms a triangle of area \(54\sqrt{3}\) square units with the coordinate axes, and where the perpendicular from the origin to the line makes an angle of \(60^\circ\) with the X-axis, we can follow these steps:

### Step 1: Understand the Geometry
The line intersects the X-axis at point \(A(a, 0)\) and the Y-axis at point \(B(0, b)\). The area of triangle \(OAB\) formed by the line and the axes is given by the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times b \]
We know the area is \(54\sqrt{3}\), so we can set up the equation:
...