Consider a one-parameter family of planes parameterized by a as in 0 = F(x, y,z, a) = ax cos a + ay.

Consider a one-parameter family of planes
parameterized by α as in 0 = F(x, y,z, α) = ax cos α + ay sin
α + bz (19.120) where a2 + b2 = 1 and constants a, b are positive and
nonzero.

(a) Show that these planes all pass through
the origin of coordinates.

(b) Using the methods of Section D.37, find
the curve of intersection of this family of planes for a given value of α.
Show that for z > 0 it is a line having spherical polar coordinates θ =
arctan(b/a), φ = π + α, and r varying.

(c) Find the envelope of this family of
planes. For z > 0 show that it is a right circular cone of half angle
θ.

(d) Find the curve of intersection and the
envelope for the region with z < 0.="" show="" that="" the="" full="" envelope="" consists="" of="" two="" cones,="" one="" inverted="" and="" one="" upright,="" with="" their="" common="" vertex="" at="" the="" origin="" of="" coordinates.="">