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Continuity And Differentiability

Question
CBSEENMA12034767
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If space straight x equals fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction comma space straight y equals fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction comma space prove space that space dy over dx plus straight x over straight y equals 0.

Solution
Here space space straight x equals fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction therefore space dx over dt equals fraction numerator left parenthesis 1 plus straight t squared right parenthesis begin display style straight d over dt end style left parenthesis 1 minus straight t squared right parenthesis minus left parenthesis 1 minus straight t squared right parenthesis begin display style straight d over dt end style left parenthesis 1 plus straight t squared right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis end fraction space space space space space space space space space space space space space equals fraction numerator left parenthesis 1 plus straight t squared right parenthesis left parenthesis negative 2 straight t right parenthesis minus left parenthesis 1 minus straight t squared right parenthesis left parenthesis 2 straight t right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction equals fraction numerator negative 2 straight t minus 2 straight t cubed minus 2 straight t plus 2 straight t cubed over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction therefore space dx over dt equals negative fraction numerator 4 straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction Now space space straight y equals fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction therefore space dy over dt equals fraction numerator left parenthesis 1 plus straight t squared right parenthesis.2 minus 2 straight t.2 straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction equals fraction numerator 2 plus 2 straight t squared minus 4 straight t squared over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction equals fraction numerator 2 minus 2 straight t squared over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction therefore space dy over dx equals fraction numerator 2 left parenthesis 1 plus straight t squared right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction therefore space dy over dx equals fraction numerator begin display style dy over dt end style over denominator begin display style dx over dt end style end fraction equals fraction numerator 2 left parenthesis 1 plus straight t squared right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction cross times fraction numerator left parenthesis 1 plus straight t squared right parenthesis squared over denominator negative 4 straight t squared end fraction equals negative fraction numerator 1 minus straight t squared over denominator 2 straight t end fraction straight L. straight H. straight S equals dy over dx plus straight x over straight y space space space space space space space space space space space equals negative fraction numerator 1 minus straight t squared over denominator 2 straight t end fraction plus fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction cross times fraction numerator 1 plus straight t squared over denominator 2 straight t end fraction equals negative fraction numerator 1 minus straight t squared over denominator 2 straight t end fraction plus fraction numerator 1 plus straight t squared over denominator 2 straight t end fraction equals 0 equals straight R. straight H. straight S

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