Distance between parallel lines
If S ≡ ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
represents a pair of parallel lines, then
i) h2 = ab
ii) af 2 = bg2 and
iii) the distance between the parallel lines is given by
or
Proof :
Let S = 0 represents the lines
lx + my + n1 = 0 ----- (1)
lx + my + n2 = 0 ----- (2)
∴ ( lx + my + n1) ( lx + my + n2) = ax2 + 2hxy +
by2 + 2gx + 2fy + c
Comparing both sides, we get
l2 = a, m2 = b, n1n2 = c,
2lm = 2h, l(n1 + n2) = 2g, m(n1 + n2) = 2f
⇒ h = lm, g = 
,
f = 
,
f =
i) h2 = (lm)2 = l2m2 = ab
∴ h2 = ab
iii) The distance between the parallel lines
Similarly the distance between the parallel lines =