Question

the distance between the pair of lines represented by the equation
x2-6xy+9y2+3x-9y-4=0 is

Answer 1

given, x² - 6xy + 9y² + 3x - 9y - 4 = 0
⇒ x² - 2.x.(3y) + (3y)² + 3(x - 3y) - 4 = 0
⇒ (x - 3y)² + 3(x - 3y) - 4 = 0
Let (x - 3y) = P
P² + 3P - 4 = 0
⇒P² + 4P - P - 4 = 0
⇒ P(P + 4) - (P + 4) = 0
⇒(P - 1)(P + 4) = 0
⇒ P = 1 , -4
Now, put P = (x - 3y)
e.g., we have two parallel equations x - 3y = -4 and x - 3y = 1
We know, distance between two parallel line is given by
|D₁ - D₂|/√(a² + b²) where equations are ax + by + D₁ = 0 and ax + by + D₂ = 0

∴ distance between two lines x - 3y = -4 and x - 3y = 1
= |-4 - 1|/√(1² + 3²)
= 5/√10 unit