Question

If the angle between the lines given by the equation x² - 3xy + λy² + 3x - 5y + 2 = 0, λ ≥ 0, is `tan^-1 (1/3)`, then λ = ______.

Options

• `1, 2/5`

• 10

• 2

• `2/3, 40`

Answer 1

If the angle between the lines given by the equation x² - 3xy + λy² + 3x - 5y + 2 = 0, λ ≥ 0,  `tan^-1 (1/3)`, then λ = 2.

Explanation:

Given, line x² - 3xy + λy² + 3x - 5y + 2 = 0

Here, a = 1, h = - 3/2, b = λ.

Angle between pairs of line is

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

`tan(tan^-1  1/3) = (2sqrt(9/4 - lambda))/(1 + lambda)`

`1/3 = (2sqrt(9/4 - lambda))/(1 + lambda)`

`=> lambda + 1 = 6sqrt((9 - 4lambda)/4)`

`=> (lambda + 1)^2 = 36/4(9 - 4lambda)`

⇒ λ² + 2λ + 1 = 81 - 36λ

⇒ λ² + 38λ - 80 = 0

⇒ (λ + 40)(λ - 2) = 0

∴ λ = 2