Multiple point

It refers to a point on a curve through which two or more branches of the curve pass.

Singular point 

A point on the curve at which the curve behaves in an extraordinary manner is called a singular point.
        There are two types of singular points:
   1. Points of inflection 
   2. Multiple point 

Point of inflection:-A curve's Inflection point is the point at which the curve's concavity changes. A point that separates the convex part of the curve from part of the curve is called a point of inflection.

Multiple point:- A point on the curve through which more than one branches of the curve pass is called a multiple point.

Double point:- A point on the curve through which two branches of the curve pass is called a double point.

Classification of double points.

There are three kinds of double points.

1. Node

A node is a point in the curve through which two real branches of the curve pass and two tangents at which are real and distinct. Thus P is a node.

2. Cusp

A double point on the curve through which two real branches of the curve pass and thw tangent at which are real and coincident is called cusp. Thus P is a cusp.

3. Conjugate point

A conjugate point on a curve is ap point in tbe neighbourhood of which there are no other real point of the curve.
        The two tangent at a conjugate point are in general imaginary but sometimes they may be real.

Note 1. Conjugate point is also called isolated points.

Note 2. The determination of the nature of double point depends basically on the nature of the two branches of the curve passing through it and not on the tangents to the curve at that point. Generally when the tangent at a double point are real, the branches are also real. But there are cases, when the tangent may be real, yet the branches may be imaginary.

☆Working rule for finding the nature of origin  which is a double point

Find the tangent at the origin by equating to zero the lowest degree terms in x and y of the equation of the curve. If the origin is a double point, then we shall get two tangents which may be real or imaginary.

      1. If two tangent are imaginary, then origin is a conjugate point.

      2. If two tangent are real and coincident, then origin is a cusp or a conjugate point.

      3. If two tangent are real and distinct, then origin is a node or a conjugate point.

To be sure, examine the nature of curve in the neighbourhood of origin. If the curve has real branches through the origin, then it is a node, otherwise a conjugate point.

Example

Q1. Examine the nature of origin for the curve :
Sol.
       The equation of curve is

Equation to zero, the lowest degree terms, the tangent at the origin are given by

 
There are two real and coincident tangent at the origin
Origin is either a cusp or a conjugate point

To be sure, we study the nature of given curve near the origin.

Now:

Now, for small value of   is negative and so y is imaginary in the neighbourhood of origin.

Origin is a conjugate point.

☆Working rule for finding the position and nature of double points of the curve

Step1. Find 

Step2. Solve the equation   to get possible double points
      Reject those points which do not satisfied the equation f (x,y)=0 of the curve remaining are the double points.

Step3. At each double point, calculate 
 (a) If D is positive, double point is a node or conjugate point.
 (b) If D=0, double point is a cusp or conjugate point.

In these cases (a) and (b), find the nature by shifting the origin to the double points and then testing the nature of tangent and existence of the curve in the neighbourhood of new origin.

 (c) If D is negative, double point is a conjugate point.

Example

Q2. Find the position and nature of the double points on the curve 
Sol.
      The equation of curve is      (1)
                     

      
 For the double points    
Now  
And
The possible double points are 
Now   does not satisfy (1)

(2,1) is the only double point
   

  

There is a node at the point (2,1).