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Question
Show that the pair of straight lines 4x2 + 12xy + 9y2 – 6x – 9y + 2 = 0 represents two parallel straight lines and also find the separate equations of the straight lines.
Solution
The given equation is 4x2 + 12xy + 9y2 – 6x – 9y + 2 = 0
Here a = 4, 2h = 12, (or) h = 6 and b = 9
h2 – ab = 62 – 4 × 9 = 36 – 36 = 0
∴ The given equation represents a pair of parallel straight lines
Consider 4x2 + 12xy + 9y2 = (2x)2 + 12xy + (3y)2
= (2x)2 + 2(2x)(3y) + (3y)2
= (2x + 3y)2
Here we have repeated factors.
Now consider, 4x2 + 12xy + 9y2 – 6x – 9y + 2 = 0
(2x + 3y)2 – 3(2x + 3y) + 2 = 0
t2 – 3t + 2 = 0 where t = 2x + 3y
(t – 1)(t – 2) = 0
(2x + 3y – 1) (2x + 3y – 2) = 0
∴ Separate equations are 2x + 3y – 1 = 0, 2x + 3y – 2 = 0
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