If lx+my+n=0 is an equation of the line joining the extremities of a pair of semi-conjugate diameters of t the ellipsex29+y24=1, then  9l2+4m2n2 is equal to

Let the extremities of a pair of semi-conjugatediameters of the ellipse be P(3cos⁡ θ, 2sin⁡ θ) Q(3cos⁡ α, 2sin⁡ α), Cbeing the centre.Then slope of CP× slope ofCQ=−49⇒23tan⁡ θ×23tan⁡ α=−49⇒1+tan⁡ θtan⁡ α=0⇒α−θ=π/2⇒ coordinates of Q are (−3sin⁡ θ, 2cos⁡ θ)equation of PQ is2(sin⁡ θ−cos⁡ θ)x−3(cos⁡ θ+sin⁡ θ)y+6=0which represents lx+my+n=0, then  l2(sin⁡ θ−cos⁡ θ)=m−3(cos⁡ θ+sin⁡ θ)=n6 So 9l2+4m2n2=(sin⁡ θ−cos⁡ θ)2+(cos⁡ θ+sin⁡ θ)2=2.