The normal at an end of a latus rectum of the ellipse x2a2+y2b2=1 passes th

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The normal at an end of a latus rectum of the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

passes through an end of the minor axis if

Options:

A . e4+e2=1

B . e3+e2=1

C . e2+e=1

D . e3+e=1

Answer: Option A
:
A
Given ellipse equation is

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

Let

P (a e, \frac{b^{2}}{a})

be one end of latus rectum.
Slope of normal at

P (a e, \frac{b^{2}}{a}) = \frac{1}{e}

Equation of normal is

y - \frac{b^{2}}{a} = \frac{1}{e} (x - a e)

It passes through

´ B (0, b)

then

b - \frac{b^{2}}{a} = - a

a^{2} - b^{2} = - a b

a^{4} e^{4} = a^{2} b^{2}

e^{4} + e^{2} = 1

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