Show that 9x2 + 24xy + 16y2 + 21x + 28y + 6 = 0 represents a pair of p

1.	Show that 9x2 + 24xy + 16y2 + 21x + 28y + 6 = 0 represents a pair of parallel straight lines and find the distance between them.
Answer» 9x² + 24xy + 16y² + 21x + 28y + 6 = 0 Here a = 9.6, b = 16, g = 21/2, f = 14, c = 6, h = 12 h² – ab = (12)² – 9(16) = 144 – 144 = 0 ∴ The lines are parallel. 9x² + 24xy + 16y² = (3x + 4y)(3x + 4y) Let 9x² + 24xy + 16y² + 21x + 28y + 6 = (3x + 4y + l)(3x + 4y + m) Equating the coefficients of x and constant term 3l + 3m = 21 lm = 6 Solving we get, l = 1 or 6 m = 6 or 1 ∴ The separate equations are 3x + 4y + 1 = 0 and 3x + 4y + 6 = 0 The distance between the parallel lines are \|(6 - 1)/√(9 + 16)\| = 5/5 = 1 unit.

Show that 9x2 + 24xy + 16y2 + 21x + 28y + 6 = 0 represents a pair of parallel straight lines and find the distance between them.

Answer»

9x² + 24xy + 16y² + 21x + 28y + 6 = 0

Here a = 9.6,

b = 16,

g = 21/2, f = 14, c = 6, h = 12

h² – ab = (12)² – 9(16) = 144 – 144 = 0

∴ The lines are parallel.

9x² + 24xy + 16y² = (3x + 4y)(3x + 4y)

Let 9x² + 24xy + 16y² + 21x + 28y + 6 = (3x + 4y + l)(3x + 4y + m)

Equating the coefficients of x and constant term

3l + 3m = 21

lm = 6

Solving we get, l = 1 or 6 m = 6 or 1

∴ The separate equations are 3x + 4y + 1 = 0 and 3x + 4y + 6 = 0

The distance between the parallel lines are |(6 - 1)/√(9 + 16)| = 5/5 = 1 unit.