Question

Show that the following equation represents a pair of line. Find the acute angle between them:

2x² + xy - y² + x + 4y - 3 = 0

Answer 1

Comparing this equation with

ax² + 2hxy + by² + 2gx + 2fy + c = 0, we get,

a = 2, h = `1/2`, b = - 1, g = `1/2`, f = 2 and c = -3

Now, abc + 2fgh - af² - bg² - ch² = 0

`=2(-1)(-3)+2(2)(1/2)(1/2)-2(2)^2-(-1)(1/2)^2-(-3)(1/2)^2`

`=6+1-8+1/4+3/4`

`=7-8+1`

`=8-8=0`

And, `h^2-ab=(1/2)^2-(2)(-1)`

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

∴ the given equation represents a pair of lines.

Let θ be the acute angle between the lines.

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

`= |(2sqrt((9/4)))/(2 - 1)|`

`= |2 xx3/2| = 3`

∴ tan θ = tan 3

∴ θ = tan^-1 (3)