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Application Of Derivatives

Question
CBSEENMA12035112

Show that the normal at any point 0 to the curve
x = a cosө + a ө sinө. y = a sin ө – a ө cos ө
is at a constant distance from the origin.

Solution

The equation of the curve are 
straight x equals straight a space cos space straight theta space plus space straight a space straight theta space sin space straight theta comma space space space space space straight y equals straight a space sin space straight theta space minus space straight a space straight theta space cos space straight theta
therefore space space space space dx over dθ space equals space minus straight a space sin space straight theta space space plus space straight a space straight theta space cos space straight theta space plus space straight a space sin space straight theta space equals space straight a space straight theta space cos space straight theta
space space space space space space space dy over dθ space equals space straight a space cos space straight theta space plus space straight a space straight theta space sin space straight theta space minus space straight a space cos space straight theta space equals space straight a space straight theta space sin space straight theta
therefore space space space space dy over dx space equals space fraction numerator begin display style dy over dθ end style over denominator begin display style dx over dθ end style end fraction space equals space fraction numerator straight a space straight theta space sin space straight theta over denominator straight a space straight theta space cos space straight theta end fraction space equals space tan space straight theta comma space space which space is space slope space of space tangent
therefore space space space space space slope space of space normal space space equals space minus fraction numerator 1 over denominator tan space straight theta end fraction space equals space minus fraction numerator cos space straight theta over denominator sin space straight theta end fraction comma
The space equation space of space normal space at space apostrophe straight theta apostrophe space is
space space space space space space space straight y minus left parenthesis straight a space sin space straight theta space minus space space straight a space straight theta space cos space straight theta right parenthesis space equals space minus fraction numerator cos space straight theta over denominator sin space straight theta end fraction left square bracket straight x minus left parenthesis straight a space cosθ space plus space straight a space straight theta space sin space straight theta right parenthesis right square bracket
space or space space space space straight y space sin space straight theta space minus space straight a space sin squared straight theta space plus space straight a space straight theta space sin space straight theta space cos space straight theta space equals space minus straight x space cos space straight theta space plus space straight a space cos squared straight theta space plus space straight a space straight theta space sin space straight theta space cos space straight theta
or space space space space space space straight x space cosθ space plus space straight y space sin space straight theta space minus space straight a left parenthesis sin squared straight theta space plus space cos squared straight theta right parenthesis space equals space 0
or space space space space space space space straight x space cos space straight theta space plus space straight y space sin space straight theta space space minus straight a space equals space 0
Its space distance space from space origin space space equals space fraction numerator open vertical bar 0 left parenthesis cos space straight theta right parenthesis space plus space 0 left parenthesis sin space straight theta right parenthesis space minus straight a close vertical bar over denominator square root of cos squared straight theta space plus space sin squared straight theta end root end fraction space equals space straight a over 1 space equals space straight a comma space which space is space constant.

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