Given the equations:
ax + by = c
bx - ay = c
We notice that the above are equations of two lines.
We need to find the relation between the lines (i.e perpendicular, parallel, or neither).
To determine the relations, we will rewrite the equations into the slope form " y= mx + a) where m is the slope.
==> ax + by = c
==> by = -ax + c
==> y (-a/b) x + c/b...............(1)
Then slope for equation (1) is m1 = -a/b.
Now we will rewrite the second equation.
==> bx -ay = c
==> -ay = -bx + c
==> y= (b/a)x - c/a..............(2)
The slope for equation (2) is m2= b/a
Now we notice that m1 and m2 are NOT equal, then they are not parallel.
However, m1*m2 = -a/b * b/a = -1
Then, the relationship between the lines is that they are perpendicular.