Find the distance between parallel lines

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

Given:

The parallel lines are 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0.

By using the formula,

The distance (d) between parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is given by

\( d= \dfrac{|C_1-C_2|}{\sqrt{A^2+B^2}} \)

= \( d= \dfrac{|-34-31|}{\sqrt{15^2+8^2}} \)

= \( d= \dfrac{|-65|}{\sqrt{225+64}} \)

= \( d= \dfrac{65}{\sqrt{289}} \)

The distance between parallel lines is \( \dfrac{65}{17}\)

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

Given:

The parallel lines are l (x + y) + p = 0 and l (x + y) – r = 0.

lx + ly + p = 0 and lx + ly – r = 0

by using the formula,

The distance (d) between parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is given by

\( d= \dfrac{|p-(-r)|}{\sqrt{l^2+l^2}} \)

= \( \dfrac{|p+r|}{\sqrt{2l}} \)

= \( \dfrac{|p+r|}{l\sqrt{2}} \)

The distance between parallel lines is \( \dfrac{|p+r|}{l\sqrt{2}} \)