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.

Answer 1

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