Question

Show that the equation 9x² – 24xy + 16y² – 12x + 16y – 12 = 0 represents a pair of parallel lines. Find the distance between them

Answer 1

The given equation of the pair of straight line is

9x² – 24xy + by² – 12x + 16y – 12 = 0  .......(1)

9x² – 24xy + 16y² = 9x² – 12xy – 12xy + 16y²

= 3x(3x – 4y) – 4y(3x – 4y)

= (3x – 4y)(3x – 4y)

Let the separate equation of the straight lines be

3x – 4y + 1 = 0 and 3x – 4y + m = 0

9x² – 24xy + 16y² – 12x + 16y – 12

= (3x – 4y + l)(3x – 4y + m)

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

3l + 3m = – 12

l + m = – 4  .......(2)

– 4l – 4m = 16

l + m = – 4  .......(3)

lm = – 12   .......(4)

(l – m)² = (l + m)² – 4lm

= (– 4)² – 4 × – 12

= 16 + 48 = 64

l – m = `sqrt(64)` = 8

l – m = 8   .......(5)

Solving equations (2) and (5), we have

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

l = `4/2`

2) ⇒ 2 + m = – 4 ⇒ m = – 6

∴ l = 2 and m = – 6

∴ The separate equation of the straight lines are

3x – 4y – 6 = 0 and 3x – 4y + 2 = 0

The distance between the parallel lines is given by

D = `(2 - ( - 6))/sqrt(3^2 + (- 4)^2`

= `(2 + 6)/sqrt(9 + 16)`

D = `8/sqrt(25)`

= `8/5`

∴ The given pair of straight lines are parallel and the distance between them is `8/5` units