Question

Show that the following equation represent a pair of lines.

Find the acute angle between them :

(x – 3)² + (x – 3)(y – 4) – 2(y – 4)² = 0.

Answer 1

Put x – 3 = X and y – 4 = Y in the given equation, we get,

X² + XY – 2Y² = 0

Comparing this equation with ax² + 2hxy + by² = 0, we get,

a = 1, h = 1/2, b = -2

This is the homogeneous equation of second degree and h² – ab = (1/2)² – 1(-2)

= 1/4 + 2 = 9/4 >0

 Hence, it represents a pair of lines passing through the new origin (3, 4).

Let θ be the acute angle between the lines.