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    Maths-
    General
    Easy

    Question

    The equation of normal to y=x log x is parallel to 2x-2y+3=0 then the equation of the normal is

    1. x plus y equals 3 e to the power of negative 2 end exponent    
    2. x minus y equals 3 e to the power of negative 2 end exponent    
    3. x minus y equals 3 e to the power of 2 end exponent    
    4. x plus y equals 3 e to the power of 2 end exponent    

    hintHint:

    A normal in geometry is a piece of geometry that is perpendicular to another piece of geometry, such as a line, ray, or vector. For instance, the (infinite) line perpendicular to the tangent line to the curve at the point is the normal line to a plane curve at the point. Here we have to find the equation of the normal. 

    The correct answer is: x minus y equals 3 e to the power of negative 2 end exponent

      At a specific point on a curve, tangents and normals behave just like any other straight lines. The normal runs perpendicular to the curve, and a tangent is parallel to the curve at the point. Like any other straight line, the equation of tangent and normal can be evaluated.
      Here we have given equation of normal to y = x log x is parallel to 2x-2y+3=0.
      N o w space d i f f e r e n t i a t i o n space t h e space g i v e n space e q u a t i o n space w i t h space r e s p e c t space t o space x comma space w e space g e t colon space space fraction numerator d y over denominator d x end fraction equals log space x plus 1 m left parenthesis tan g e n t right parenthesis equals log x plus 1 N o w space w e space k n o w space t h a t space n o r m a l space i s space a l w a y s space p e r p e n d i c u l a r space t o space tan g e n t space s o space w e space g e t colon m 1 cross times m 1 equals negative 1 S o space w e space h a v e colon m left parenthesis n o r m a l right parenthesis equals negative fraction numerator 1 over denominator log x plus 1 end fraction E q u a t i o n space o f space n o r m a l space i s space g i v e n space b y colon y minus y 1 equals m left parenthesis x minus x 1 right parenthesis N o w space w e space w i l l space c o m p a r e space t h e space s l o p e s space o f space n o r m a l comma space w e space g e t colon space m equals 1 minus fraction numerator 1 over denominator log x plus 1 end fraction equals 1 x equals 1 over e squared N o w space t h a t space t h e space p o i n t space l i e s space o n space c u r v e comma space l e t s space s u b s t i t u t e space x space a n d space f i n d space y comma space w e space g e t colon y equals negative 2 over e squared S o space t h e space e q u a t i o n space o f space n o r m a l space w i l l space b e colon y plus 2 over e squared equals x minus 1 over e squared y e squared plus 2 equals x e squared minus 1 x e squared minus y e squared equals 3 x minus y equals 3 e to the power of negative 2 end exponent

      So here we used the concept of tangents and normals, we found the slope and then went to proceed for for the final answer. Using equation of tangent we found equation of the normal is x minus y equals 3 e to the power of negative 2 end exponent.

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