Explicit Cartesian Equations :-
If A 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 t 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 .
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