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Easy

Question

The equation of the one of the tangents to the curve y equals cos invisible function application left parenthesis x plus y right parenthesis comma blank minus 2 pi less or equal than x less or equal than 2 pi that is parallel to the line x plus 2 y equals 0, is

  1. x plus 2 blank y equals 1  
  2. x plus 2 blank y equals fraction numerator pi over denominator 2 end fraction  
  3. x plus 2 blank y equals fraction numerator pi over denominator 4 end fraction  
  4. None of these  

The correct answer is: x plus 2 blank y equals fraction numerator pi over denominator 2 end fraction


    We have,

    y equals cos invisible function application left parenthesis x plus y right parenthesis

    rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative sin invisible function application open parentheses x plus y close parentheses open parentheses 1 plus fraction numerator d y over denominator d x end fraction close parentheses …(i)

    Since the tangent is parallel to x plus 2 blank y equals 0

    therefore Slope of the tangent equals negative fraction numerator 1 over denominator 2 end fraction

    rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative fraction numerator 1 over denominator 2 end fraction

    Putting fraction numerator d y over denominator d x end fraction equals negative fraction numerator 1 over denominator 2 end fraction in (i), we get

    negative fraction numerator 1 over denominator 2 end fraction equals negative sin invisible function application open parentheses x plus y close parentheses open parentheses 1 minus fraction numerator 1 over denominator 2 end fraction close parentheses

    rightwards double arrow sin invisible function application left parenthesis x plus y right parenthesis equals 1 ….(ii)

    Now,

    y equals cos invisible function application open parentheses x plus y close parentheses and 1 equals sin invisible function application left parenthesis x plus y right parenthesis

    rightwards double arrow y to the power of 2 end exponent plus 1 equals 1 rightwards double arrow y to the power of 2 end exponent equals 0 rightwards double arrow y equals 0

    Putting y equals 0 in y equals cos invisible function application left parenthesis x plus y right parenthesis and sin invisible function application left parenthesis x plus y right parenthesis equals 1, we get

    sin invisible function application x equals 1 and cos invisible function application x equals 0 rightwards double arrow x equals negative fraction numerator pi over denominator 2 end fraction comma blank minus fraction numerator 3 pi over denominator 2 end fraction

    Thus, the points on the curve y equals cos invisible function application left parenthesis x plus y right parenthesis where tangents are parallel to x plus 2 y equals 0 are open parentheses pi divided by 2 comma blank 0 close parentheses comma blank open parentheses negative 3 pi divided by 2 comma blank 0 close parentheses

    The equation of the tangent at open parentheses pi divided by 2 comma blank 0 close parentheses is

    y minus 0 equals negative fraction numerator 1 over denominator 2 end fraction open parentheses x minus fraction numerator pi over denominator 2 end fraction close parentheses rightwards double arrow x plus 2 y equals fraction numerator pi over denominator 2 end fraction

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