# 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. Let$m_1 \, \, and \, \, m_2$ 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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