Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 16] Envelope 233
Solution. The given family of curves depends on the parameter a. Form
the system of equations (1):
J *cosa + y sin a p = 0,
\ x sin a + y cos a = 0.
Solving the system for x and y t we obtain parametric equations of the
envelope
x = pcosa, r/ =
Squaring both equations and adding, we eliminate the parameter a:
frig. 82
Thus, the envelope of this family of straight lines is a circle of radius p
with centre at the origin. This particular family of straight lines is a family
of tangent lines to this circle (Fig. 82).
2063. Find the envelope of the family of circles
2064. Find the envelope of the family of straight lines
(k is a variable parameter).
2065. Find the envelope of a family of
radius R whose centres lie on the jc-axis.
circles of the same
2066. Find a curve which forms an envelope of a section
of / length when its end-points slide along the coordinate axes.
2067. Find the envelope of a family of straight lines that
form with the coordinate axes a triangle of constant area S.
2068. Find the envelope of ellipses of constant area S whose
axes of symmetry coincide.