# If sinα , sinβ and cosα are in G.P, then roots of the equation x^2 + 2x cot β + 1 = 0 are always.

Q: If sinα , sinβ and cosα are in G.P, then roots of the equation x2 + 2x cot β + 1 = 0 are always.

(A) equal

(B) real

(C) imaginary

(D) greater than 1

Sol. sinα , sinβ , cosα are in G.P.

⇒ sin2β = sinα cosα

⇒ cos2β = 1 – sin2α ≥ 0

Now, the discriminant of the given equation is

4cot2β – 4 = 4 cos2β × cosec2β ≥ 0

⇒ Roots are always real.

Hence (B) is the correct answer.