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If the normals at an end of a latus rect...

If the normals at an end of a latus rectum of an ellipse passes through the other end of the minor axis, then prove that `e^(4) + e^(2) =1.`

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The equation of the minor of an end `L(ae, (b^(2))/(a))` of a latus rectum of the ellipse `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) =1` is given by
`(x-ae)/((ae)/(a^(2)))=(y-(b^(2))/(a))/((b^(2))/(ab^(2)))`
`rArry-(b^(2))/(a)=(1)/(e)(x-ae)`
`rArr ay -b^(2) = (ax)/(e) -a^(2)`
which will pass through `B'(0,-b)` if
`-ab - b^(2) = 0 - a^(2)`
`rArr ab = a^(2) -b^(2)`
`rArr aasqrt(1-e^(2)) =a^(2) -a^(2)(1-e^(2)) =a^(2)e^(2)`
`rArr 1 - e^(2) = e^(4)`
`rArr e^(4) + e^(2) =1`
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