**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 .