Question

If m₁ and m₂ are roots of the equation `x^2 + (sqrt(3) + 2)x + (sqrt(3) - 1)` = 0 then the area of the triangle formed by the lines y = m₁x, y = m₂x and y = c is ______.

Options

• `[(sqrt(33) - sqrt(11))/4]c^2`

• `[(sqrt(33) + sqrt(11))/4]c^2`

• `[(sqrt(33) + sqrt(11))/2]c^2`

• `[(sqrt(33) - sqrt(11))/2]c^2`

Answer 1

If m₁ and m₂ are roots of the equation `x^2 + (sqrt(3) + 2)x + (sqrt(3) - 1)` = 0 then the area of the triangle formed by the lines y = m₁x, y = m₂x and y = c is `underlinebb([(sqrt(33) + sqrt(11))/4]c^2)`.

Explanation:

m₁ and m₂ are roots of `x^2 + (sqrt(3) + 2)x + (sqrt(3) - 1)` = 0

m₁ + m₂ = `-(sqrt(3) + 2)`

m₁m₂ = `(sqrt(3) - 2)`

Area = `1/2|(0, c/m_1, c/m_2),(0, c, c),(1, 1, 1)|`

= `1/2[c^2/m_1 - c^2/m_2]`

= `1/2c^2[(m_2 - m_1)/(m_1m_2)]`

= `c^2/2[sqrt((m_1 + m_2)^2 - 4m_1m_2)/(m_1m_2)]`

= `c^2/2[sqrt((sqrt(3) + 2)^2 - 4(sqrt(3) - 1))/(sqrt(3) - 1)]`

= `c^2/2[sqrt(3 + 4 + 4sqrt(3) - 4sqrt(3) + 4)/(sqrt(3) - 1)]`

= `c^2/2sqrt(11)/(sqrt(3) - 1) xx (sqrt(3) + 1)/(sqrt(3) + 1)`

= `c^2/2.(sqrt(33) + sqrt(11))/2`