The locus of the extremities of the latus rectum of the family of ellipses b²x²+y²=a²b^{2 having a given major} axes is :

Hello student,
The given ellipse is b²x² + y² = a²b²
i.e. x²/a² + y²/b² = 1.
(ab)² = a²(1 – e²) or b² = 1 - e².
Let an extremity of latus rectum be L(x, y).
Then, x = ae gives x² = a²e² = a²(1 – b²) .... (a)
and y = (ab)²/a = ab² .... (b)
From (a) and (b) we have x² = a² – a(ab²)
Hence, locus of L or L’ = x² =a² ± ay.