Question

Find the equation of the tangent and the normal to the curve $straight x to the power of 2 over 3 end exponent plus straight y to the power of 2 over 3 end exponent space equals space 2 space space at space left parenthesis 1 comma space 1 right parenthesis$

Solution

The equation of curve is
$straight x to the power of 2 over 3 end exponent plus straight y to the power of 2 over 3 end exponent space equals space 2$
Differentiating both sides w.r.t.x, we get.
$2 over 3 straight x to the power of negative 1 third end exponent plus 2 over 3 straight y to the power of negative 1 third end exponent dy over dx space equals space 0 space space space space space space space or space space space space straight y to the power of negative 1 third end exponent dy over dx space equals space minus straight x to the power of negative 1 third end exponent$
$therefore space space space space space space space dy over dx space equals space minus straight x to the power of negative begin display style 1 third end style end exponent over straight y to the power of negative begin display style 1 third end style end exponent space equals space straight y to the power of begin display style 1 third end style end exponent over straight x to the power of begin display style 1 third end style end exponent space equals space minus open parentheses straight y over straight x close parentheses to the power of 1 third end exponent$
At (1, 1), slope of tangent $equals negative open parentheses 1 over 1 close parentheses to the power of 1 third end exponent space equals space minus 1$
$therefore space space space space space space equation space of space tangent space at space left parenthesis 1 comma space 1 right parenthesis space is space space space space space space space space space space space space straight y space minus 1 space equals space minus left parenthesis straight x minus 1 right parenthesis comma space space space or space space space space straight y minus 1 space equals space minus straight x plus 1 space space space or space space straight x plus straight y minus 2 space equals space 0 At space left parenthesis 1 comma space 1 right parenthesis comma space slope space of space normal space space equals space minus fraction numerator 1 over denominator negative 1 end fraction space equals space 1 therefore space space equaiton space of space normal space at space left parenthesis 1 comma space 1 right parenthesis space is space space space space space space space space space space space space space space space space space straight y space minus 1 space space equals space 1 space left parenthesis straight x minus 1 right parenthesis space space space space space or space space space space space straight y space minus 1 space equals space straight x minus 1 space space space or space space straight x minus straight y space equals space 0$

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Application Of Derivatives

Find the equation of the tangent and the normal to the curve $straight x to the power of 2 over 3 end exponent plus straight y to the power of 2 over 3 end exponent space equals space 2 space space at space left parenthesis 1 comma space 1 right parenthesis$

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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