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