Question

If the equation S ≡ ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a pair of parallel straight lines, then show that 

i) h² = ab, 

ii) af² = bg² and 

iii) the distance between the parallel lines 2√(g² - ac/ a(a+b))

Answer 1

Let the two parallel straight lines represented by S = 0 be 

lx + my + n₁ = 0 —— (1) 

and lx + my + n₂ = 0. —— (2) 

Then, S ≡ λ(lx + my + n₁) (lx + my + n₂), for some real λ ≠ 0. 

From this we have

Now

so that af² = bg². 

(iii) Distance between the parallel lines