Tangents and Normals





P(x,y) is any point on the curve f(x,y)=c. PT is tangent at p and PN is normal at P. Angle made by tangent PT with x-axis is denoted by \psi in anticlockwise direction.

 

m =m tan \psi is defined as slope of gradient of tangent PT.

We also define m = tan \psi = ( \dfrac{dy}{dx} ) _{ ( x_1 , y_1 ) }

= slope of tangent,

Slope of normal = \dfrac{ - 1}{ ( \dfrac{dy}{dx} ) }

Equation of Tangent PT is y - y_1 = n ( x - x_1 )

 

Equation of normal PN is m ( y - y_1 ) + ( x - X_1 ) = 0

PM is perpendicular from P on x-axis.

By \Delta PMT ,

 

\dfrac{PM}{TM} = tan \psi \leftarrow TM = y_1 cot \psi

 

By \Delta PMN , \dfrac{PM}{MN} = tan( < PNT ) = tan ( 90 - \psi ) = cot \psi

\rightarrow MN = y_1 tan \psi

 

We define:

Sub tangent = TM = y_1 cot \psi

Sub normal = MN = y_1 tan \psi

Length of Tangent = PT = y_1 cosec \psi

Length of normal = PN = y_1 sec \psi

 

Where P ( x_1 , y_1 ) is point P.

The tangent is parallel to x-axis if:

 

\psi = o \, \, or \, \, if \dfrac{dy}{dx} = 0

 

The tangent is parallel to y –axis if:

 

\psi = \dfrac{ \pi}{2} \, \, or \, \, if \dfrac{dx}{dy} = 0

 

Important Note:

Tangent at the origin is obtained by equating to zero the lowest degree terms, provided the curve passes through origin.

 

 

Definition of Angle of Intersection

 

 

Suppose two curves f_1 = c_1 \, \, and \, \, f-2 = c_2 cut at P. Letm_1 \, \, and \, \, m_2 be gradient of the two tangents to the two curves at the point of inserction. Angle \theta between the two curves at P is defined as angle \theta between the two tangents at P.

The two curves cut orthogonally if m_1 m_2 = - 1



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