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Question

The equation of the normal at the point (2, 3) on the ellipse 9 x to the power of 2 end exponent plus 16 y to the power of 2 end exponent equals 180, is

  1. 3 y equals 8 x minus 10    
  2. 3 y minus 8 x plus 7 equals 0    
  3. 8 y plus 3 x plus 7 equals 0    
  4. 3 x plus 2 y plus 7 equals 0    

The correct answer is: 3 y minus 8 x plus 7 equals 0


    fraction numerator x minus x subscript 1 end subscript over denominator x subscript 1 end subscript divided by a to the power of 2 end exponent end fraction equals fraction numerator y minus y subscript 1 end subscript over denominator y subscript 1 end subscript divided by b to the power of 2 end exponent end fraction, which is the standard equation of normal at point left parenthesis x subscript 1 end subscript comma y subscript 1 end subscript right parenthesis.
    In the given ellipse, a to the power of 2 end exponent equals 20 comma b to the power of 2 end exponent equals fraction numerator 180 over denominator 16 end fraction.
    Hence the equation of normal at the point left parenthesis 23 right parenthesis is
    fraction numerator x minus 2 over denominator 2 divided by 20 end fraction equals fraction numerator y minus 3 over denominator 48 divided by 180 end fraction Þ 40 left parenthesis x minus 2 right parenthesis equals 15 left parenthesis y minus 3 right parenthesis
    Þ 8 x minus 3 y equals 7Þ 3 y minus 8 x plus 7 equals 0.

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