Tracing of curves

Tracing of Curves

    Introduction

        In this chapter we shall learn the methods of tracing a curve in general and the                    properties of some standard curves commonly met in engineering problems. 

      Some Important point for Tracing the curves

     1. Symmetry

  Find out if the curve is symmetrical about any line.

·  If the equation of the curve remains unchanged when y is

    replaced by- y i.e. if the equation contains only even powers

    of y.  Then the curve is symmetrical about the x-axis.

·  If the equation of the curve remains unchanged when x is replaced by-x i.e. if the equation contains only even powers of x. Then the curve is symmetrical about y-axis.

·  If the equation of the curve remains unchanged when both x and y are replaced by- x and - y. Then the curve is symmetrical in opposite quadrants.

·  If the equation of the curve remains unchanged when x and y are interchanged (i.e. put x=y). Then the curve is symmetrical about the line x=y.

2.  Origin

           If x=0 and y=0 (i.e. origin (0,0)) satisfies the given equation of curve then we can say that curve passes through the origin ( i.e. the curve passes through the origin if there is no constant terms in the equation of curve )

3. Equation of Tangents

           When curve passes through the origin then find the equation of tangents at origin by equating to zero the lowest degree terms.

(i).  If there is two real and distinct tangents ( i.e. non coincident tangent ) then origin is a node.

(ii)  If there is two real and equal tangents ( i.e. coincident tangent ) then origin is a cusp.

(iii) If there is two imaginary tangents then origin is conjugate or isolated point.

(iv) If there is only one tangent, then origin is a point of inflection.

4. Point of Intersection of curves with coordinates axes

       Puts x=0 in the equation of curve we get the value of y for the points, were the curve
            cuts the y-axis.

            Similarly put y=0 in the equation of curve we get the value of x for the points, were
            the curve cuts the x-axis.

5.    Asymptotes

A straight line at a finite distance from origin to which the curve does not intersect it and touches it at infinity, it means that the asymptotes is a boundary line or the limiting line for the curve.

          • Asymptotes parallel to y-axis

            Equating the coefficients  of highest power of x to zero in given equation of curve.

        • Asymptotes parallel to x-axis

           Equating the coefficient of highest power of  y to zero in the given equation of the
           curve

     NOTE-It is not necessary that every curve have Asymptotes.

6.  Region

                To find the region of the curve we solve the equation of the curve for the y or x, which

                 ever convenience.

                Suppose we solve the equation of curve for y, then we examine fallowing points

·         We find those values of x for which y tends to infinite

·         We find the interval for x in which the value of y become imaginary.

·         We find the interval for x in which the value of y either increasing or decreasing.

7.   Special point

                     Find dy/dx by the equation of curve at some value of x.

   ·      If dy/dx is zero for some value of x then the tangent is parallel to x-axis.

   ·      If dy/dx is infinite for some value of x then tangents is parallel to y-axis.

   ·      If dy/dx is positive for some interval of x then curve is increasing in this interval.

   ·      If dy/dx is negative for some interval of x then curve is decreasing.

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Singular Point

Singular Point 

Definition 

A point on a curve at which the curve exhibits an unusual behaviour is called a Singular point ( like as point of maxima and minima )

Type of Singular Point 

The two main type of singular point is given below.

(I)   Multiple point 

(II)  Point of Inflexion 

Multiple point

A point on curve through which more than one branch of curve passes, is called a multiple point of curve .

If two branches of the curve passes through a point, then point on the is known as Double point.

Thus a curve has two tangents at a double point, one for each branch. And these tangents may be real and distinct, real and coincident or imaginary.

Type of Double point

There are three types of double point.

(i)   Node

A double point on a curve through which pass two real and distinct tangent, is known as a node.

(ii)  Cusp 

A double point on a curve through which pass two real and coincident tangent, is known as a cusp.

 

(iii) Conjugate point or Isolated point 

If there are no real points on the curve in the neighbourhood of a point R on the curve then R is called a conjugate point or isolated point

Point of Inflexion 

A point where the curve crosses the tangents is called point of inflexion  

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