lfthe lines `a_1x + b_ 1y + c_1 = 0` and `a_2x + b_2y + c_2 = 0` are perpendicular, then

`m_1m_2 =-1` `=> (-a_1/b_1) xx ( - a_2/b_2) = -1`

`=> a_1 a_2 +b_1 b_2 =0`

lffollows from the above discussion that the lines `a_ 1x + b_1y+c_1 = 0` and `a_2x + b_2y+c_2 = 0` are

(1) Condition if `a_1/a_2 = b_1/b_2 = c_1/c_2`

(2) Parallel if `a_1/a_2 = b_1/b_2 ne c_1/c_2`

(3) Interesecting if `a_1/a_2 ne b_1/b_2`

(4) Perpendicular if `a_1a_2 +b_1b_2 =0`

The equation of a line parallel to a given line `ax + by + c = 0` is
`ax+by+lambda = 0`
where `A` is a constant and value of `A` can be determined using another given condition

The equation of a line perpendicular to a given line `ax+ by+ c = 0` is
`bx - ay+ lambda = 0`
where `A` is a constant and value of `A` can be determined using another given condition.

Let `(x_1 , y_1)` be any point on the line `ax + by +c_2 = 0`

Distance of point `(x_1, y_1)` from the line `ax+ by +c_1 = 0` is

`p=(|ax_1+by_1+c|)/(sqrt(a^2+b^2))`

Now point `(x_1, y_1)` lies on `ax + by+c_2 = 0` then

`ax_1 +by_1 +c_2 = 0`

`=> ax_1 + by_1 = -c_2`

`=> p=(|c_1-c_2|)/(sqrt(a^2+b^2))`