Course Outline

&        Envelope and its examples.

&        Envelope of straight lines.

&        Envelope of a family of curves.

&        Envelope of two parametric family of curves.

Introduction

Envelope

If each of the members of the family of curves C º F(x, y, a) = 0 touches a fixed curve E, then E is called the envelope of the family of curves C. The curve E also, at each point, is touched by the member of the family C.

Envelope of a Straight Line

The equation of envelope of the family of straight lines F(x, y, a)
º y – f(a) x – f (a) = 0 (a being the parameter) is the a- eliminant of F = 0 and
dF/dx = 0

Envelope of Curve

If a curve E exists, which touches each member of a family of curves
C [º f(x, y, a) = 0], the curve E is called the envelope of C. Since E is the locus of the points of contact of family of curves f(x, y, a) = 0 the point where the curve f(x, y, a) = 0 for a particular value of a touches E, depends upon that value of a.