Question

Show that the equation 4x² + 4xy + y² – 6x – 3y – 4 = 0 represents a pair of parallel lines. Find the distance between them

Answer 1

The given equation of pair of straight lines is

4x² + 4xy + y² – 6x – 3y – 4 = 0   ......(1)

4x² + 4xy + y² = (2x + y)²

Let the separate equation of the lines be

2x + y + l = 0   ......(2)

2x + y + m = 0   ......(3)

4x² + 4xy + y² – 6x – 3y – 4 = (2x + y + l)(2x + y + m)

Comparing the coefficients of x, y and constant terms on both sides we have

2l + 2m = – 6

l + m = – 3   ......(4)

l + m = – 3   ......(5)

l m = – 4   ......(6)

(l – m)² = (l + m)² – 4lm

(l – m )² = (– 3)² – 4 × – 4

(l – m)² = 9 + 16 = 25

l – m = 5 ………… (7)

Solving equations (4) and (7)

(4) ⇒   l +  m = – 3
(7) ⇒   l –  m =     5
         2l +  0 =      2 

⇒ l = 1

4) ⇒ l + m = – 3

⇒ m = – 4

∴ The separate equation of the straight lines are

2x + y + 1 =0 and 2x + y – 4 = 0

The distance between the parallel lines is

D = `(- 4 - 1)/sqrt(2^2 + 1^2)`

= `(- 5)/sqrt(4 + 1)`

D = `5/sqrt(5)`

= `sqrt(5)`

∴ The given equation represents a pair of parallel straight lines and the distance between the parallel lines is `sqrt(5)` units.