Question

Let − π/6 < θ < π/12. Suppose α₁ and β₁ are the roots of the equation x² − 2xsecθ + 1 = 0 and α₂ and β₂ are the roots of the equation x² + 2xtanθ − 1 = 0. If α₁ > β₁ and α₂ > β₂, then α₁ + β₂ equals

(a)   2(secθ  − tanθ )

(b)   2secθ 

(c)  −2tanθ 

(D)  0

Answer 1

Correct option  (c)  −2tanθ

We have θ  (-π/6,π/12).

• It is given that α₁ and β₁ are the roots of the equation

⇒ α_{1 ,}β₂ = secθ ± tanθ (since secθ > 0 and tanθ < 0)

Since it is given that α_{1 >} β₂, we get

α₁= secθ − tanθ

• It is also given that α_2 and β₂ are the roots of the equation

x² + 2x tan − 1 = 0

Since it is given that α_2 and β_2, we get

α₁ = tanθ +  secθ

and  β₂ =  tanθ -  secθ

Therefore,

α₂ + β₂ = secθ - tanθ -  secθ =  -2tanθ