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