The value of a has to be determined for which the lines a^2x - 3y + 8 = 0 and 3x + ay + 9 = 0 are parallel.
Parallel lines have equal slope. Express the two lines in the slope intercept form, y = mx + b
a^2x - 3y + 8 = 0
=> 3y = a^2x + 8
=> `y = (a^2/3)x + 8/3`
The slope of this line is `a^2/3`
3x + ay + 9 = 0
=> -ay = 3x + 9
=> `y = (-3/a)x - 9/a`
The slope of this line is `-3/a`
Equating the slope of the lines `a^2/3 = -3/a`
=> `a^3 = -9`
`a = -root(3) 9 ~~ -2.08`
The value of a for which the lines are parallel is `-root(3) 9` .
This value can be negative. The slope of a line can be of either sign, a negative value indicates a decrease in the value of y for an increase in the value of x.