Applications of the Derivative to Geometry 6-51
6.7 ENVELOPES
Consider the equation of the curve
f x y x y( , ) , = + = 0
which is a straight line. At times we also encounter the equations of curves such as
x a y a acos sin ,+ =
which is also a straight line. However, it should be noted that the above equation represents a straight
line for every value of a. So, choosing different values of a, we can obtain an infinite number of
straight lines all which are equidistant from the origin. Similarly, the equation represents a series of
parabolas with vertex (h, k) and axis y = k but different latus rectum, for a being arbitrary and k, h
being fixed.
( ) ( ).y k a x h− = −
2
4
So, we may say that the equation of a curve f(x, y) = 0 also ...