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If a and b are positive numbers such that a greater than b comma then the minimum value of a   sec invisible function application blank theta minus b tan invisible function application theta open parentheses 0 less than theta less than fraction numerator pi over denominator 2 end fraction close parentheses is

  1. fraction numerator 1 over denominator square root of a to the power of 2 end exponent minus b to the power of 2 end exponent end root end fraction  
  2. fraction numerator 1 over denominator square root of a to the power of 2 end exponent plus b to the power of 2 end exponent end root end fraction  
  3. square root of a to the power of 2 end exponent plus b to the power of 2 end exponent end root  
  4. square root of a to the power of 2 end exponent minus b to the power of 2 end exponent end root  

The correct answer is: square root of a to the power of 2 end exponent minus b to the power of 2 end exponent end root


    Let y equals a sectheta minus btantheta
    rightwards double arrow blank fraction numerator d y over denominator d theta end fraction equals a blank s e c theta t a n theta minus b s e c to the power of 2 end exponent theta

    Put fraction numerator d y over denominator d theta end fraction equals 0 rightwards double arrow s e c theta open parentheses a blank t a n theta minus b s e c theta close parentheses equals 0

    rightwards double arrow s i n theta equals fraction numerator b over denominator a end fraction(because s e c theta not equal to 0)

    Now, fraction numerator d to the power of 2 end exponent y over denominator d theta to the power of 2 end exponent end fraction greater than 0 comma a t blank s i n theta equals fraction numerator b over denominator a end fraction

    thereforeminimum value is

    y equals a fraction numerator a over denominator square root of a to the power of 2 end exponent minus b to the power of 2 end exponent end root end fraction minus b fraction numerator b over denominator square root of a to the power of 2 end exponent minus b to the power of 2 end exponent end root end fraction

    =square root of a to the power of 2 end exponent minus b to the power of 2 end exponent end root

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