If the normal at the point P(θ) to the ellipse x214+y5=1 intersects it again at the point Q(2θ), then cos⁡θ is equal to

The normal at P(acos⁡θ,bsin⁡θ) is axcos⁡θ−bxsin⁡θ=a2−b2, where a2=14,b2=5 It meets the curve again at Q(2θ) , i.e., (acos⁡2θ,bsin⁡2θ)∴ acos⁡θ(acos⁡2θ)−bsin⁡θ(bsin⁡2θ)=a2−b2⇒ 14cos⁡θ(cos⁡2θ)−5sin⁡θ(sin⁡2θ)=14−5⇒ 28cos2⁡θ−14−10cos2⁡θ=9cos⁡θ⇒ 18cos2⁡θ−9cos⁡θ−14=0⇒ (6cos⁡θ−7)(3cos⁡θ+2)=0⇒cos⁡θ=-23