Multiple points

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

. 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

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

but sometimes they may be real.





Curve tracing

1. Symmetry

includes techniques that can be used to draw a rough idea of overall shape of a plane curve. For curve sketching we need to perform certain steps on the given equation of the curve.For curve sketching we need to understand certain termsA point through which two or more than two branches of a curve pass is called a multiple point.In particular, if two branches of a curve pass through a point, the point is called a double point.We shall keep in mind the following point for tracing the graph of the equation Curve given by is symmetric aboutx-axis if it unchanged on changing y to -y i.e, if