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Monday, 23 December 2013

Equations of Tangent and Normal

 

Explicit Cartesian Equations :-

If      be the angle which the tangent at any point   (x, y)   on the curve     y = f (x)     makes with    x   axis then ;
                tan A = dy/dx = f' (x) 

Therefore , the equation of the tangent at any point     (x , y)     on the curve    y = f (x)      is

              Y - y = f' (x) (X - x)  -------------(1)

where   X , Y   are the current co-ordinates of any point on the tangent .

The normal to the curve    y = f (x)    at any point    (x , y)     is the straight line which  passes through that point ans is perpendicular to the tangent to the curve at the point so that its slope is ;

             -1/f (x)

Hence the equation of the normal at   (x , y)    to the curve    y= f (x)    is ;

            (X - x) + f' (x) (Y - y) = 0 

Implicit Cartesian Equations :-

If any point    (x , y) , then the curve   f (x, y) = 0

          Where   Dy/Dx    is not equivalent to  0   .

           dy/dx =  - (Df/Dx) / (Df/Dy)

Hence the equations of the tangent and the normal at any point
(x , y)   on the curve    f (x , y) = 0  are ;

         (X - x)(Df / Dx) + (Y - y) (Df / Dy) = 0      and 

        (X - x) (Df / Dy) - (Y - y)(Df / Dx) = 0

Parametric Cartesian Equations  :- 

At the pont  of the curve    x = f (t) , y = F(t)  ;
       where we have  f'(t)   is not equivalent to   0    ;
we have ;

    dy/dx = (dy/dt) (dt/dx) = F' (t)/f' (t)  

Hence the equations of the tangents and the normal at any point    t    of the curve    x=f(t) , y=F(t)   are ;

   [X-f(t)]F'(t)-[Y-F(t)]f'(t)=0
   [X-f(t)]f'(t)+[Y-F(t)]F'(t)=0

respectively .