Application of Derivatives

Find the equation of the tangent and normal to the curve y 2 = 4ax at (at 2 , 2at) - Wired Faculty

Question

Find the equation of the tangent and normal to the curve y² = 4ax at (at², 2at)

Solution

The equation of curve is $straight y squared space equals space 4 ax$
$therefore space space space space space 2 straight y dy over dx space equals space 4 straight a space space space space space space space space space space space space space space space rightwards double arrow space space space space dy over dx space equals space fraction numerator 2 straight a over denominator straight y end fraction$
$At space left parenthesis at squared comma space space 2 at right parenthesis comma space space space dy over dx space equals space fraction numerator 2 straight a over denominator 2 at end fraction space equals space 1 over straight t$
$therefore space space space space space space space at space left parenthesis at squared comma space space 2 at right parenthesis comma space space slope space of space tangent space space equals space 1 over straight t therefore space space space space space space equation space of space tangent space to space the space curve space at space left parenthesis at squared comma space space space 2 at right parenthesis space is space space space space space space space space space space space space straight y minus 2 at space equals space 1 over straight t left parenthesis straight x minus at squared right parenthesis space space space space or space space space space space ty minus 2 at squared space equals space straight x minus at squared space space or space space space ty space equals space straight x plus at squared. At space left parenthesis at squared comma space space 2 at right parenthesis comma space space space slope space of space normal space equals space space straight t therefore space space space space space space equation space of space normal space to space the space curve space at space left parenthesis at squared comma space space space 2 at right parenthesis space is space space space space space space space space space space space straight y minus 2 at space equals space minus straight t left parenthesis straight x minus at squared right parenthesis space or space space space space space space space straight y minus 2 at space equals space minus tx plus at cubed or space space space space space space space straight y space equals negative tx plus 2 at plus at cubed. space space$

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