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Vector Algebra

Question
CBSEENMA12032806

Find the area of smaller region bounded by the ellipse straight x squared over straight a squared plus straight y squared over straight b squared space equals 1 and the straight line straight x over straight a plus straight y over straight b equals 1

Solution

Consider the equations
                 straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1                             ...(1)
and       straight x over straight a plus straight y over straight b equals 1                                       ...(2)

Line (2) meets ellipse (1) in A (a, 0) and B (0, b)
Area of shaded region = Area of region OAB area of ∆OAB
                                equals space straight b over straight a integral subscript 0 superscript straight a square root of straight a squared minus straight x squared end root space dx minus straight b over straight a integral subscript 0 superscript straight a left parenthesis straight a minus straight x right parenthesis space dx
equals space straight b over straight a open square brackets fraction numerator straight x square root of straight a squared minus straight x squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent straight x over straight a close square brackets subscript 0 superscript straight a minus space fraction numerator straight b over denominator 2 straight a end fraction open square brackets fraction numerator left parenthesis straight a minus straight x right parenthesis squared over denominator 1 end fraction close square brackets subscript 0 superscript straight a
equals space straight b over straight a open square brackets open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 1 close parentheses minus open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 0 close parentheses close square brackets plus fraction numerator straight b over denominator 2 straight a end fraction left square bracket 0 minus straight a squared right square bracket
equals space straight b over straight a open square brackets straight a squared over 2 cross times straight pi over 2 minus 0 minus 0 close square brackets space minus space ab over 2 space equals space πab over 4 minus ab over 2 space equals space ab over 4 left parenthesis straight pi minus 2 right parenthesis space sq. space units.
which is required area. 

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