Question

The normal at an end of a latus rectum of the ellipse `x^2/a^2 + y^2/b^2 = 1` passes through an end of the minor axis if (A) `e^4+e^2=1` (B) `e^3+e^2=1` (C) `e^2+e=1` (D) `e^3+e=1`

Answer 1

Normal at L:
`(a^2x)/x_1-(b^2y)/y_1=a^2-b^2`
`(a^2x)/(ap)-(b^2y)/(b^2/a)=a^2-b^2`
(-b,a)lies on line
`0+ab=a^2-b^2`
`(ab)^2=(a^2-b^2)`
`a^2-a^2(1-b^2)=(a^2-a^2(1-e^2))`
`a^4(1-p^2)=a^4e^2`
`1=e^2+e^4`.