Application of Derivatives

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

Question

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

Solution

The equation of curve is y² = 4ax
$therefore space space space space space 2 straight y dy over dx space equals 4 straight a space space space space space rightwards double arrow space space space space space space space dy over dx space equals space fraction numerator 2 straight a over denominator straight y end fraction$
$At space left parenthesis at squared comma space space 2 at right parenthesis comma space space space dy over dx equals fraction numerator 2 straight a over denominator 2 at end fraction space equals space 1 over straight t therefore space space space space space space at space left parenthesis at squared comma space space 2 at right parenthesis comma space space slope space of space tangent space equals space 1 over straight t therefore space space space space equation space of space tangent space to space the space curve space at space left parenthesis at squared comma space space space 2 at right parenthesis space is space space space space space space space space space space space space space space straight y minus 2 at space equals space 1 over straight t left parenthesis straight x minus at squared right parenthesis space space space or space space space space space ty minus 2 at squared space equals space straight x minus at squared space space or space space space ty space equals space straight x plus at squared At space left parenthesis at squared comma space space 2 at right parenthesis comma space slope space of space normal space equals space minus straight t therefore space space space equation space of space normal space to space the space curve space at space left parenthesis at squared comma space space space 2 at right parenthesis space is space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 2 at space equals space minus straight t left parenthesis straight x minus at squared right parenthesis. or space space space space space space space space straight y minus 2 at space equals space minus tx plus at cubed or space space space space space space space space straight y space equals space minus tx plus 2 at plus at cubed.$

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