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Application Of Derivatives

Question
CBSEENMA12035137
Wired Faculty App

Find the equation of the tangent and normal to the given curves at the points given:
straight y squared space equals space fraction numerator straight x cubed over denominator 4 minus straight x end fraction space space at space space left parenthesis 2 comma space space minus 2 right parenthesis

Solution

The equation of the curve is straight y squared space equals space fraction numerator straight x cubed over denominator 4 minus straight x end fraction
Differentiating both sides w.r.t x,
                    2 straight y dy over dx space equals space fraction numerator left parenthesis 4 minus straight x right parenthesis. space 3 straight x squared minus space straight x cubed left parenthesis negative 1 right parenthesis over denominator left parenthesis 4 minus straight x right parenthesis squared end fraction
therefore space space space space space 2 straight y dy over dx space equals space fraction numerator 12 straight x squared minus 3 straight x cubed plus straight x cubed over denominator left parenthesis 4 minus straight x right parenthesis squared end fraction space space space space space rightwards double arrow space space space space space dy over dx space equals space fraction numerator 12 straight x squared minus 2 straight x cubed over denominator 2 straight y left parenthesis 4 minus straight x right parenthesis squared end fraction
     At space left parenthesis 2 comma space minus 2 right parenthesis space comma space dy over dx space equals space fraction numerator 12 left parenthesis 2 right parenthesis squared minus 2 left parenthesis 2 right parenthesis cubed over denominator 2 left parenthesis negative 2 right parenthesis thin space left parenthesis 4 minus 2 right parenthesis squared end fraction space equals space fraction numerator 48 minus 16 over denominator negative 4 space cross times space 4 end fraction space equals space fraction numerator 32 over denominator negative 16 end fraction space equals space minus 2 comma which space is space slope space of space tangent therefore space space space space space the space equation space of space tangent space at space left parenthesis 2 comma space minus 2 right parenthesis space is space space space space space space space space space space space space space space space space space space space straight y minus left parenthesis negative 2 right parenthesis space equals space minus 2 left parenthesis straight x minus 2 right parenthesis or space space space space straight y plus 2 space equals space minus 2 straight x plus 4 space space space space space or space space space 2 straight x plus straight y minus 2 space equals space 0 Sloe space of space normal space equals space 1 half therefore space space space space the space equation space of space normal space at space left parenthesis 2 comma space minus 2 right parenthesis space is space space space space space space space space space space space space space space space space straight y minus left parenthesis negative 2 right parenthesis space equals space 1 half left parenthesis straight x minus 2 right parenthesis comma space space space space space space or space space space straight y plus 2 space equals space 1 half left parenthesis straight x minus 2 right parenthesis or space space space space space space 2 straight y plus 4 space equals space straight x minus 2 space space space space space space space space or space space space straight x minus 2 straight y minus 6 space equals space 0

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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