The solution of the system of equations x + cos ⁡ α y = 1 and cos ⁡ α x + 4 y = 2 satisfy x ≥ 4 5 and y ≤ 1 2 , then the value of α can lie in the interval

Step-by-step solution

Δ = 1 cos ⁡ α cos ⁡ α 4 = 4 - c o s 2 α Δ 1 = 1 cos ⁡ α 2 4 = 2 2 - cos ⁡ α Δ 2 = 1 1 cos ⁡ α 2 = 2 - cos ⁡ α ∵   x = 2 2 + cos ⁡ α ≥ 4 5 ⇒ 5 ≥ 2 cos ⁡ α + 4 ⇒ cos ⁡ α ≤ 1 2 ∵   y = 1 2 + cos ⁡ α ≤ 1 2 ⇒ 2 + cos ⁡ α ≥ 2 ⇒ cos ⁡ α ≥ 0 ⇒ α ∈ π 3 , π 2