Maths-
General
Easy

Question

If x equals e to the power of t end exponent sin invisible function application t comma blank y equals e to the power of t end exponent cos invisible function application t comma blank t h e n fraction numerator d to the power of 2 end exponent y over denominator d x to the power of 2 end exponent end fraction blank a t blank x equals pi i s

  1. 2 e to the power of pi    
  2. 1 half e to the power of pi    
  3. fraction numerator 1 over denominator 2 e to the power of pi end fraction    
  4. 2 over e to the power of pi    

The correct answer is: 2 over e to the power of pi


    Since, x equals e to the power of t end exponent sin invisible function application t a n d blank y equals e to the power of t end exponent cos invisible function application t
    rightwards double arrow blank fraction numerator d x over denominator d t end fraction equals e to the power of t end exponent cos invisible function application t plus sin invisible function application t e to the power of t end exponent
    and fraction numerator d y over denominator d t end fraction equals blank minus e to the power of t end exponent sin invisible function application t plus e to the power of t end exponent cos invisible function application t
    therefore blank fraction numerator d y over denominator d x end fraction equals fraction numerator d y divided by d t over denominator d x divided by d t end fraction equals fraction numerator left parenthesis cos invisible function application t minus sin invisible function application t right parenthesis over denominator left parenthesis cos invisible function application t plus sin invisible function application t right parenthesis end fraction
    fraction numerator d to the power of 2 end exponent y over denominator d x to the power of 2 end exponent end fraction equals blank fraction numerator open square brackets table row cell left parenthesis cos invisible function application t plus sin invisible function application t right parenthesis left parenthesis negative sin invisible function application t minus cos invisible function application t right parenthesis end cell row cell negative left parenthesis cos invisible function application t minus sin invisible function application t right parenthesis left parenthesis negative sin invisible function application t plus cos invisible function application t right parenthesis end cell end table close square brackets over denominator left parenthesis cos invisible function application t plus sin invisible function application t right parenthesis to the power of 2 end exponent end fraction fraction numerator d t over denominator d x end fraction
    equals fraction numerator negative left parenthesis sin invisible function application t plus cos invisible function application t right parenthesis to the power of 2 end exponent minus left parenthesis cos invisible function application t minus sin invisible function application t right parenthesis to the power of 2 end exponent over denominator left parenthesis cos invisible function application t plus sin invisible function application t right parenthesis to the power of 2 end exponent end fraction
    cross times fraction numerator 1 over denominator e to the power of t end exponent left parenthesis cos invisible function application t plus sin invisible function application t right parenthesis end fraction
    equals negative fraction numerator 2 over denominator e to the power of t end exponent left parenthesis cos invisible function application t plus sin invisible function application t right parenthesis to the power of 3 end exponent end fraction
    rightwards double arrow open parentheses fraction numerator d to the power of 2 end exponent y over denominator d x to the power of 2 end exponent end fraction close parentheses subscript left parenthesis x equals pi right parenthesis end subscript equals fraction numerator negative 2 over denominator e to the power of pi end exponent left parenthesis cos invisible function application pi plus sin invisible function application pi right parenthesis to the power of 3 end exponent end fraction equals fraction numerator 2 over denominator e to the power of pi end exponent end fraction

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