If the normal at the point P(θ) to the ellipse x214+y25=1intersect it again at the point Q(2θ), then cos⁡ θ=

Normal at P(14cos⁡ θ,5sin⁡ θ) to the ellipse is14xcos⁡ θ−5ysin⁡ θ=14−5=9Since it passes through Q(2θ)14×14cos⁡ 2θcos⁡ θ−5×5sin⁡ 2θsin⁡ θ=9⇒142cos2⁡ θ−1cos⁡ θ−5×2sin⁡ θcos⁡ θsin⁡ θ=9⇒18cos2⁡ θ−9cos⁡ θ−14=0⇒(3cos⁡ θ+2)(6cos⁡θ−7)=0⇒cos⁡ θ=−2/3.