If P is the length of the perpendicular from a focus upon the tangent at any point P of the ellipsex2/a2+y2/b2=1 and  r r is the distance of P from the focus,  then 2ar−b2p2=

The equation of the tangent at P(acos⁡θbsin⁡θ)t   to the ellipse  x2/a2+y2/b2=1  isxacos⁡θ+ybsin⁡θ=1 length of the perpendicular from the focus (ae, 0) on the line is p=ecos⁡θ−1cos2⁡θa2+sin2⁡θb2=ab(ecos⁡θ−1)b2cos2⁡θ+a21−cos2⁡θ=ab(ecos⁡θ−1)a2−a2e2cos2⁡θ=b1−ecos⁡θ1+ecos⁡θb2p2=1+ecos⁡θ1−ecos⁡θ Now,   r2=(ae−acos⁡θ)2+b2sin2⁡θ=a2(e−cos⁡θ)2+1−e2sin2⁡θ=a2e2cos2⁡θ−2ecos⁡θ+1=a2(1−ecos⁡θ)2 ⇒          r=a(1−ecos⁡θ)  Now 2ar−b2p2=21−ecos⁡θ−1+ecos⁡θ1−ecos⁡θ=1