Lecture 17 MATH 121 (Wg Cdr Monir)

 MATH 121 (Wg Cdr Monir) 

Tangent and Normal

1. The tangent at P to a given curve is defined as the limiting position of the secant PQ (when such a limit exists) as the point Q approaches P along the curve.

    Derivative of any point on a curve is the slope of the tangent at the point (x,y) on the curve.
That is \(\frac{dy}{dx} = \tan \psi = \) slope of the tangent.
In a curve \(y = f(x)\), if at a particular point (x,y)
    1.  \(\frac{dy}{dx} = 1\), Then the slope of the tangent at that point of the curve makes an angle of \(45^\circ\) with the positive direction of x-axis.
    2. \(\frac{dy}{dx}\) = Negative, Then the tangent to the curve at that point makes an obtuse angle with the positive direction of x-axis.
    3.  \(\frac{dy}{dx} = 0\) , Then the tangent to the curve at that point makes an angle of \(0^\circ\) with the x-axis, i.e the tangent to the curve at that point is parallel to the x-axis.
    4. \(\frac{dy}{dx} = \infty\), then the tangent to the curve at that point makes on angle of \(\frac{\pi}{2}\) with the x-axis, i.e the tangent to the curve at the point is perpendicular to x-axis.

    Equation of tangent at any point (x,y) of the curve  \(y = f(x)\)
  $$ Y - y = \frac{dy}{dx} (X - x) $$  

    Equation of the tangent at any point on the curve \(f(x,y) = 0\)  $$ (X - x) f_x + (Y - y) f_y = 0 $$  

If the curve be given by the algebraic equation  \(f(x,y) = 0\) which can be made homoheneous in x,y and z the equation of the tangent will be  \(X f_x + Y f_y + Z f_x = 0\)  

    Equation of the tangent at any point on the curve \(f(x,y) = 0\)
if the equation of the curve be given in the form  \(f(x,y) = 0\) then  \(\frac{dy}{dx} = \frac{ - \frac{\partial f}{\partial x}}{\frac{\partial f }{\partial y}}\)  
substituting this expression for \(\frac{dy}{dx}\) in  \begin{align*} & Y - y = \frac{dy}{dx} (X - x)\\ \textrm{we have} &, \\ & Y - y = \frac{ - \frac{\partial f}{\partial x}}{\frac{\partial f }{\partial y}} (X - x)\\ or, ~ & (X - x) \frac{\partial f}{\partial x} + (Y - y) \frac{ \partial f}{\partial y} = 0 \end{align*}  
as the equation of the tangent.
Replacing \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) by \(f_x\) and \(f_y\) respectively we may write the equation of the tangent in the form  \((X - x) f_x + (Y - y) f_y = 0\)  

    ♦ Normal and its equation.
Normal: Normal to a curve at a point is defined as the straight line through that point at right angles to the tangent to the curve at that point.
We know that tangent to a curve  \( y = f(x) ~ \textrm{at} ~ (x,y) ~\textrm{is} \)  \(Y - y = \frac{dy}{dx} (X - x) \tag{1}\label{eq:1}\)  
any line throught(x,y) is given by  \(Y - y = m (X - x) \tag{2}\label{eq:2}\)
If this line is normal to the curve at (x,y) this is perpendicular to the tangent to the curve, i.e 

  \begin{align*} & m \frac{dy}{dx} = - 1\\ \therefore ~& m = - \frac{1}{\frac{dy}{dx}} \end{align*}  
From (2) the equation of the normal at (x,) to the curve  \(y = f(x)\) is  \begin{align*} & Y - y = - \frac{1}{\frac{dy}{dx}} (X - x) \\ or, ~ & (Y - y ) \frac{dy}{dx} + (X - x) = 0 \end{align*}