The equations of the two straight lines through (7,9) and making an angle of 60degree with the line x-3^1/2y-2*3^1/2=0, are:

Let the slope of the line be .

The given line can be written as  which is of the form y = mx + c.

Slope of the given line = .

It is given that the angle between the required line and line  is 60°.

We know that if θ is the acute angle between lines l₁ and l₂ with slopes m₁ and m₂ respectively then 

. Hence,

The first equation gives
But the second equation does not seem to be true unless .
Hence, the two required slopes are .
Equation of a line passing through (7, 9) and slope
 .... (1st equation)

Equation of a line passing through (7, 9) and slope  is

x = 7              ....(2nd equation)


ALTERNATE:
Since the given equation has a slop of , it must be making an angle of 30 degree with the x axis. Any line making 60 degree of angle with this line will effectively make an angle of 30 + 60 = 90 degree or 30 - 60 = -30 degree (or 150 degree) with the x-axis.
Now, 90 degree means slope of and 150 degree means slope of .