It is known that the distance (d) between parallel lines \(Ax + By + C1 = 0\) and \(Ax + By + C2 = 0 \) is given by
\(d=\frac{|C_1-C_2|}{\sqrt{A^2+B^2}}\)
(i) The given parallel lines are \(15x + 8y - 34 = 0\) and \(15x + 8y + 31 = 0.\)
Here, \(A = 15, B = 8, C1 = -34\), and \(C2 = 31\).
Therefore, the distance between the parallel lines is
\(d=\frac{|C_1-C_2|}{\sqrt{A^2+B^2}}\)
\(=\frac{|-34-31|}{\sqrt{(15)^2+(8)^2}}\) units
\(=\frac{|-65|}{17}\) units
\(=\frac{65}{17} \) units.
(ii) The given parallel lines are \( l (x + y) + p = 0\) and \(l (x + y) - r = 0.\)
\(lx + ly + p = 0\) and \( lx + ly - r = 0 \)
Here, \(A = l, B = l, C_1 = p\), and \(C_2 = - r.\)
Therefore, the distance between the parallel lines is
\(d=\frac{|C_1-C_2|}{\sqrt{A^2+B^2}}\)
\(=\frac{|p+r|}{\sqrt{l^2+l^2}}\) units
\(=\frac{|p+r|}{\sqrt{2l^2}}\) units
\(=\frac{|p+r|}{l\sqrt{2}} \) units
\(=\frac{1}{\sqrt{2}}|\frac{p+r}{l}|\) units.