Question

Show that the equation 2x² + xy - y² + x + 4y - 3 = 0 represents a pair of lines. Also, find the acute angle between them.

Answer 1

Comparing the equation

2x² + xy - y² + x + 4y - 3 = 0 with

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

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

We have condition,

abc + 2fgh − af² − bg² − ch²

= `(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`

= − 1 + 1

= 0

∴ abc + 2fgh − af² − bg² − ch² = 0  ....(i)

Also, h² - ab = `(1/2)^2 - 2(- 1)`

= `1/4 + 2`

= `9/4 > 0`

∴ h² - ab > 0   ...(ii)

From (i) and (ii) given equation represents pair of lines.

Let θ be the acute angle between the lines

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

`= |(2sqrt((1/2)^2 - 2(-1)))/(2-1)|`

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

`= |(2sqrt9/4)/1|`

`= |2 xx 3/2|`

= |3|

∴ θ = tan^-1(3)