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6. The tangent at a point P (a cos ϕ, b sin ϕ) of an ellipse  x 2 a 2 + y 2 b 2 = 1 ,  meets its auxiliary circle in two points, the chord joining which subtends a right angle at the centre. Show that the eccentricity of the ellipse is 1 + sin 2   ϕ - 1 2 .

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x squared plus y squared equals r squared T h e space c h o r d space w h i c h space s u b t e n d s space 90 to the power of ring operator space a t space t h e space c e n t r e space i s space a t space a space d i s tan c e space o f fraction numerator r over denominator square root of 2 end fraction space f r o m space t h e space c e n t r e E q u a t i o n space o f space tan g e n t space i s space fraction numerator x cos empty set over denominator a end fraction plus fraction numerator y sin empty set over denominator b end fraction equals 1 D i s tan c e space f r o m space left parenthesis 0 comma 0 right parenthesis equals square root of fraction numerator a squared plus b squared over denominator 2 end fraction end root equals fraction numerator 1 over denominator square root of begin display style fraction numerator cos squared empty set over denominator a squared end fraction end style plus begin display style fraction numerator sin squared empty set over denominator b squared end fraction end style end root end fraction S o space w e space h a v e fraction numerator a squared b squared over denominator b squared cos squared empty set plus a squared sin squared empty set end fraction equals fraction numerator a squared plus b squared over denominator 2 end fraction b squared minus b squared sin squared empty set plus a squared sin squared empty set equals fraction numerator 2 a squared b squared over denominator a squared plus b squared end fraction sin squared empty set open parentheses a squared minus b squared close parentheses plus b squared equals fraction numerator 2 a squared b squared over denominator a squared plus b squared end fraction sin squared empty set open parentheses fraction numerator a squared minus b squared over denominator b squared end fraction close parentheses plus 1 equals fraction numerator 2 a squared over denominator a squared plus b squared end fraction N o w space w e space k n o w space t h a t space e squared equals 1 minus b squared over a squared S o space t h a t space a squared over b squared equals fraction numerator 1 over denominator 1 minus e squared end fraction T h e space a b o v e space e q u a t i o n space b e c o m e s sin squared empty set open parentheses fraction numerator 1 over denominator 1 minus e squared end fraction minus 1 close parentheses plus 1 equals fraction numerator 2 over denominator 1 plus 1 minus e squared end fraction sin squared empty set open parentheses fraction numerator e squared over denominator 1 minus e squared end fraction close parentheses equals fraction numerator e squared over denominator 2 minus e squared end fraction sin squared empty set fraction numerator 1 over denominator 1 minus e squared end fraction equals fraction numerator 1 over denominator 2 minus e squared end fraction open parentheses 2 minus e squared close parentheses sin squared empty set equals 1 minus e squared e squared open parentheses 1 minus sin squared empty set close parentheses equals 1 minus 2 sin squared empty set e squared equals fraction numerator cos squared empty set minus sin squared empty set over denominator cos squared empty set end fraction e equals square root of 1 minus tan squared empty set end root L o o k s space l i k e space t h e r e space i s space s o m e space e r r o r space i n space q u e s t i o n R e g a r d s
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