Question

If $straight x equals straight a space sin space 2 straight t left parenthesis 1 plus cos space 2 straight t right parenthesis space and space straight y space equals straight b space cos space 2 straight t left parenthesis 1 minus cos space 2 straight t right parenthesis comma space then space find dy over dx space at space straight t space equals straight pi over 4.$

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Solution

straight x equals straight a space sin space 2 straight t left parenthesis 1 plus cos space 2 straight t right parenthesis comma straight y equals space straight b space cos space 2 straight t left parenthesis 1 minus cos space 2 straight t right parenthesis dx over dt space equals 2 straight a space cos space 2 straight t left parenthesis 1 plus cos space 2 straight t right parenthesis space plus space straight a space sin space 2 straight t left parenthesis negative 2 space sin space 2 straight t right parenthesis space space space space equals space 2 straight a space cos 2 straight t space plus space 2 acos squared 2 straight t minus 2 straight a space sin squared 2 straight t space space space space equals 2 acos 2 straight t space plus space 2 acos 4 straight t space space space space dy over dt equals negative 2 bsin 2 straight t left parenthesis 1 minus cost space 2 straight t right parenthesis plus bcos 2 straight t left parenthesis 2 sin 2 straight t right parenthesis space space space space space space space equals space minus 2 bsin 2 straight t space plus 2 bsin space 2 tcos 2 straight t space plus 2 bcos 2 straight t space sin 2 straight t space space space space space space space equals negative 2 bsin 2 straight t plus 4 bsin 2 tcos 2 straight t space space space space space space space equals negative 2 bsin 2 straight t plus 2 bsin 4 straight t

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 negative 2 bsin 2 straight t plus 2 bsin 4 straight t over denominator 2 acos 2 straight t plus 2 acos 4 straight t end fraction space space space rightwards double arrow space space space dy over dx equals fraction numerator negative 2 bsin 2 straight t plus 2 bsin 4 straight t over denominator 2 acos 2 straight t plus 2 acos 4 straight t end fraction space space space space space rightwards double arrow space open vertical bar dy over dx close vertical bar subscript straight t equals straight pi over 4 end subscript space equals space fraction numerator negative 2 bsin begin display style fraction numerator 2 straight pi over denominator 4 end fraction end style plus 2 bsin begin display style fraction numerator 4 straight pi over denominator 4 end fraction end style over denominator 2 acos begin display style fraction numerator 2 straight pi over denominator 4 end fraction end style plus 2 acos begin display style fraction numerator 4 straight pi over denominator 4 end fraction end style end fraction space space space space space rightwards double arrow open vertical bar dy over dx close vertical bar subscript straight t equals straight pi over 4 end subscript space equals fraction numerator negative 2 bsin begin display style straight pi over 2 end style plus 2 bsinπ over denominator 2 acos begin display style straight pi over 2 end style plus 2 acosπ end fraction space space space space space rightwards double arrow open vertical bar dy over dx close vertical bar subscript straight t equals straight pi over 4 end subscript space equals fraction numerator negative 2 straight b over denominator negative 2 straight a end fraction equals straight b over straight a space space space space space space space space therefore space space space open vertical bar dy over dx close vertical bar subscript straight t equals straight pi over 4 end subscript space equals straight b over straight a space space space space space space space space space

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Vector Algebra

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0

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NCERT Sample Papers

Entrance Exams Preparation

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Vector Algebra

If

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:y' + y = ex

Determine order and degree (if defined) of differential equations:y'' + (y')2 + 2 y = 0

Determine order and degree (if defined) of differential equations:y″ + 2y′ + sin y = 0

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0