The normal at a point P on the ellipse x2+4y2=16 meets  the x -axis at Q . If M is the midpoint of the line segment PQ , then the locus of M intersects the latus rectums of the  given ellipse at points

Normal at P is given by 4xsec⁡ϕ−2ycosec⁡ϕ=12∴Q≡(3cos⁡ϕ,0) Let mid-point of PQ be M(α,β) . ∴α=3cos⁡ϕ+4cos⁡ϕ2=72cos⁡ϕ or cos⁡ϕ=27α and β=sin⁡ϕ Using cos2⁡ϕ+sin2⁡ϕ=1 , we have 449α2+β2=1 or 449x2+y2=1----(1) Now, the latus rectum to above ellipse is x=±23-----(2) Solving (1) and (2), we have 4849+y2=1 or y=±17 The points of intersection are (±23,±1/7) .