Question

Show that the equation 2x² − xy − 3y² − 6x + 19y − 20 = 0 represents a pair of intersecting lines. Show further that the angle between them is tan⁻¹(5)

Answer 1

TThe equation of the given pair of lines is

2x² – xy – 3y² – 6x + 19y – 20 = 0   .......(1)

Compare this equation with the equation

ax² + 2hxy + by² + 2gx + 2fy + c = 0  .......(2)

a = 2 

2h = – 1

b = – 3

2g = – 6

2f = 19

c = – 20

h² – ab = `(- 1/2)^2 - 2(2)( - 3)`

= `1/4 + 6 ≠ 0`

∴ The given line (1) is not parallel.

∴ They are intersecting lines.

Let θ be the angle between the lines.

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

tan θ = `root(2)(( - 1/2)^2 - (2)( - 3))/(2 + ( - 3))`

= `root(2)(1/4 + 6)/(- 1)`

= `- 2sqrt((1 + 24)/4`

tan θ = ` - 2/2 sqrt(25)`

= – 5

Taking the acute angle θ = tan⁻¹(5)