Question

Show that the following equation represents a pair of line. 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.

∴ tan θ = `|(2 sqrt("h"^2 - "ab"))/("a + b")|`

here a = 1, 2h = 1, i.e. h = `1/2` and b = - 2

∴ tan θ = `|(2sqrt((1/2)^2 - 1(-2)))/(1 - 2)|`

`= |(2(sqrt(1/4 + 2)))/-1|`

`= |(2 xx 3/2)/-1|`

∴ tan θ = 3

∴ θ = tan^-1(3)