The quadratic equation tanθ x2+2(secθ+cosθ)x+(tanθ+3√2cotθ) always has
complex roots for all θ
Discriminant, △=b2−4ac
=(2(secθ+cosθ))2−4tanθ(tanθ+3√2cotθ)
=4(sec2θ+cos2θ+2)−4(tan2θ+3√2)]
=4[(sec2θ+cos2θ+2−(tan2θ+3√2)]
=4[(sec2θ+tan2θ+2+cos2θ+3√2)]
=4[1+2+cos2θ−3√2)
=4[3+cos2θ−3√2]
=4[cos2θ+3−3√2]
As cos2θ≤1
So, △≤4[1+3−3√2]
=4(4−3√2)
≃4(4−3×1.4)
=4(−0.2)
⇒△ is negative
⇒No real roots