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Question

The eccentricity of an ellipse whose centre is at the origin is 1 half spaceIf one of its directrices isx equals negative 4 then the equation of the normal to it at open parentheses 1 comma fraction numerator 3 over denominator 2 end fraction close parentheses text end textis

  1. 4 x minus 2 y equals 1    
  2. 4 x plus 2 y equals 7    
  3. x plus 2 y equals 4    
  4. 2 y minus x equals 2    

The correct answer is: 4 x minus 2 y equals 1


    The given ellipse is depicted in the following figure:

    Eccentricity of ellipse is 1/2. Now
    fraction numerator negative a over denominator e end fraction equals negative 4
    That is,
    a equals 4 cross times fraction numerator 1 over denominator 2 end fraction equals 2
    Therefore,
    b to the power of 2 end exponent equals a to the power of 2 end exponent open parentheses 1 minus e to the power of 2 end exponent close parentheses equals a to the power of 2 end exponent open parentheses 1 minus fraction numerator 1 over denominator 4 end fraction close parentheses equals 3
    Hence, the equation of the given ellipse is written as
    fraction numerator x to the power of 2 end exponent over denominator 4 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 3 end fraction equals 1
    fraction numerator x over denominator 2 end fraction plus fraction numerator 2 y over denominator 3 end fraction cross times y to the power of ´ end exponent equals 0
    y to the power of ´ end exponent equals fraction numerator negative 3 x over denominator 4 y end fraction
    open y to the power of ´ end exponent close vertical bar subscript left parenthesis 13 divided by 2 right parenthesis end subscript equals fraction numerator negative 3 over denominator 4 end fraction cross times fraction numerator 2 over denominator 3 end fraction equals fraction numerator negative 1 over denominator 2 end fraction
    Therefore, the equation of normal at open parentheses 1 comma fraction numerator 3 over denominator 2 end fraction close parentheses text  is end text
    table row cell y minus fraction numerator 3 over denominator 2 end fraction equals 2 left parenthesis x minus 1 right parenthesis end cell row cell 2 y minus 3 equals 4 x minus 4 end cell row cell rightwards double arrow 4 x minus 2 y equals 1 end cell end table

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