Application of Derivatives

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Find the slope of the normal to the curve - Wired Faculty

Question

Find the slope of the normal to the curve $straight x space equals space 1 minus straight a space sin space straight theta comma space space space straight y space equals space space straight b space cos squared straight theta space space at space straight theta space equals space straight pi over 2.$

Solution

straight x space equals space 1 minus straight a space sin space straight theta comma space space space straight y space equals space space straight b space cos squared straight theta space space at space straight theta space equals space straight pi over 2.

The equations of the curve are

straight x space equals space space 1 minus straight a space sin space straight theta comma space space space space straight y space equals space straight b space cos squared straight theta

therefore space space space space space dx over dθ space equals space minus straight a space cos space straight theta comma space space dy over dθ space equals space minus 2 straight b space cos space straight theta space sin space straight theta

Now

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 negative 2 straight b space sinθ space cosθ over denominator negative straight a space cosθ end fraction space equals space fraction numerator 2 straight b over denominator straight a end fraction sin space straight theta

straight theta space equals space straight pi over 2 comma space

dy over dx space equals space fraction numerator 2 straight b over denominator straight a end fraction sin space straight pi over 2 space equals space fraction numerator 2 straight b over denominator straight a end fraction cross times 1 space equals space fraction numerator 2 straight b over denominator straight a end fraction

This is slope of tangent to the curve

therefore space space space space slope space of space normal space to space the space curve space equals space minus fraction numerator 1 over denominator begin display style fraction numerator 2 straight b over denominator straight a end fraction end style end fraction space equals space minus fraction numerator straight a over denominator 2 straight b end fraction