Question

Find the distance between parallel lines

(i) 15x + 8y - 34 = 0 and 15x + 8y + 31 = 0

(ii) l(x + y) + p = 0 and l(x + y) - r = 0

Answer 1

(i) The given equations are

15x + 8y - 34 = 0

and 15x + 8y + 31 = 0

where a = 15, b = 8, $c_1$

= - 34 and $c_2$ = 31

$\therefore$ The distance between two parallel lines

$d=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}$

$\Rightarrow \frac{|-34-31|}{\sqrt{(15{)}^2+(8{)}^2}}\Rightarrow \frac{65}{17}\text{units}$

(ii) We have the equation

lx + ly + p = 0

and lx + ly - r = 0

where a = 1, b = 1, $c_1$ = p

and $c_2=-r$

$\therefore$ The distance between two parallel lines

$\Rightarrow \frac{|p+r|}{\sqrt{l^2+l^2}}\Rightarrow \frac{1}{\sqrt{2}}\left|\frac{p+r}{l}\right|\text{units}$