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Find the range of eccentricity of the el...

Find the range of eccentricity of the ellipse `x^2/a^2+y^2/b^2=1`, (where a > b) such that the line segment joining the foci does not subtend a right angle at any point on the ellipse.

Text Solution

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Let any point P on the ellipse b `P(a cos theta,b sin theta)` .
If `angleS_(1)PS_(2)=po//2`, then P less on the circle having `S_(1)S_(2)` as its diameter
Therefore, the equation of the circle drawn on `S_(1)s_(2)` as its `x^(2)y^(2)=a^(2)e^(2)`
`a^(2)-b^(2)`
Since the point P should not lie on the ellipse, there should not be any point on the intersection of
`x^(2)+y^(2)=a^(2)-b^(2) and (x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
`:. a^(2)-b^(2)ltb^(2)`
or `2b^(2)gta^(2)`
`or 1-e^(2)gt(1)/(2)`
or `e^(2)lt(1)/(2)`
or `e in(0,(1)/(sqrt(2)))`
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