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Vector Algebra

Question
CBSEENMA12032803

Find the area of the smaller region bounded by the ellipse straight x squared over 9 plus straight y squared over 4 space equals space 1 and the straight line straight x over 3 plus straight y over 2 space equals space 1.

Solution

The equation of ellipse is
                straight x squared over 9 plus straight y squared over 4 space equals space 1                   ...(1)
 The equation of line is
              straight x over 3 plus straight y over 2 space equals space 1                       ...(2)
Let line (2) meet ellipse (1) in A (3, 0) and B(0, 2).
         Area of shaded region = Area of region OAB - area of increment OAB

equals space 2 over 3 integral subscript 0 superscript 3 square root of 9 minus straight x squared end root space dx minus space 2 over 3 integral subscript 0 superscript 3 left parenthesis 3 minus straight x right parenthesis space dx                       open square brackets because space of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis close square brackets
equals space 2 over 3 open square brackets fraction numerator straight x square root of 9 minus straight x squared end root over denominator 2 end fraction plus 9 over 2 sin to the power of negative 1 end exponent straight x over 3 close square brackets subscript 0 superscript 3 space minus space 2 over 3 open square brackets fraction numerator left parenthesis 2 minus straight x right parenthesis squared over denominator left parenthesis negative 1 right parenthesis thin space left parenthesis 2 right parenthesis end fraction close square brackets subscript 0 superscript 3
equals space 2 over 3 open square brackets open parentheses fraction numerator 3 square root of 9 minus 9 end root over denominator 2 end fraction plus 9 over 2 sin to the power of negative 1 end exponent 1 close parentheses minus open parentheses 0 plus 9 over 2 sin to the power of negative 1 end exponent 0 close parentheses close square brackets plus 1 third left square bracket 0 minus 9 right square bracket
space equals space 2 over 3 open square brackets open parentheses 0 plus 9 over 2 cross times straight pi over 2 close parentheses minus left parenthesis 0 plus 0 right parenthesis close square brackets plus 1 third left parenthesis 0 minus 9 right parenthesis
equals space 2 over 3 cross times 9 over 2 cross times straight pi over 2 minus 1 third space equals space fraction numerator 3 straight pi over denominator 2 end fraction minus 3
equals space fraction numerator 3 space straight pi minus 6 over denominator 2 end fraction space equals 3 over 2 left parenthesis straight pi minus 2 right parenthesis space sq. space units.

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