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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 dydx=tanψ= slope of the tangent.
In a curve y=f(x), if at a particular point (x,y)
    1.  dydx=1, Then the slope of the tangent at that point of the curve makes an angle of 45 with the positive direction of x-axis.
    2. dydx = Negative, Then the tangent to the curve at that point makes an obtuse angle with the positive direction of x-axis.
    3. 
 dydx=0 , Then the tangent to the curve at that point makes an angle of 0 with the x-axis, i.e the tangent to the curve at that point is parallel to the x-axis.
    4. dydx=, then the tangent to the curve at that point makes on angle of π2 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)
  Yy=dydx(Xx)  

    Equation of the tangent at any point on the curve f(x,y)=0  (Xx)fx+(Yy)fy=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  Xfx+Yfy+Zfx=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  dydx=fxfy  
substituting this expression for dydx in  Yy=dydx(Xx)we have,Yy=fxfy(Xx)or, (Xx)fx+(Yy)fy=0  
as the equation of the tangent.
Replacing fx and fy by fx and fy respectively we may write the equation of the tangent in the form  (Xx)fx+(Yy)fy=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) at (x,y) is  Yy=dydx(Xx)  
any line throught(x,y) is given by  Yy=m(Xx)
If this line is normal to the curve at (x,y) this is perpendicular to the tangent to the curve, i.e 

  mdydx=1 m=1dydx  
From (2) the equation of the normal at (x,) to the curve  y=f(x) is  Yy=1dydx(Xx)or, (Yy)dydx+(Xx)=0  

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