Question

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

Answer 1

Comparing the given equation with ax² + 2kxy + by² = 0 

We get a = 9, h = -12, b = 16. 

Now h² = (-12)² = 144, ab = (9) (16) = 144

h² = ab ⇒ The given equation represents a pair of parallel lines. 

To find their separate equations: 

9x² – 24xy + 16y² = (3x – 4y)² 

So, 9x² – 24xy + 16y² – 12x + 16y – 12 = (3x – 4y + l )(3x – 4y + m)

Here coefficient of x ⇒ 3m + 3l = -12 ⇒ m + l = -4 

Coefficient of y ⇒ -4m – 4l = 16 ⇒ m + l = -4 

Constant term l m = -12 

Now l + m = -4 and lm = -12 ⇒ l = -6 and m = 2 

So the separate equations are 3x – 4y – 6 = 0 and 3x – 4y + 2 = 0

The  distance between the parallel lines is