6.5 Translation and rotation of axes 399

have hyperbolic type? Parabolic type? Elliptic type? Hint: Get on the right

track.

13 We should know something about the possibility of obtaining information

about the graph of the equation

14x2+2Bxy+Cy2+F=O

by solving for x or y by use of the quadratic formula. Other cases being similar
or much easier, supposethat C > 0, B 0 0, and F 5,6 0. Show that a point P
lies on the graph iff (iff means if and only if) its coordinates satisfy the equation

y

-Bx ± (B2

C

• 
  

  AC)x2 - CF

  
  

Show that if B2 - 14C > 0, then the points (x,y) for which jxiis sufficiently
great and
X
Y l-± (BZ-SIC)-B zF
J

must lie on the graph, and tell why the graph cannot be an ellipse. Show that
if B2 - .IC < 0, the graph cannot be a hyperbola.
14 Sketch a graph of the equation

y=x+ x2

by sketching graphs of yl = x and y2 = 1 - x2 and adding ordinates (that is,
values of y). Give some precise information about the graph.
15 It is sometimes said that the graph of the equation

x + y3i = a%,

where a is a positive constant, is a parabola. Is this true? Ans.: No, but the
graph is a part of a parabola.
16 A wheel of radius a rolls, without slipping and with angular speed w, on
the top side of the x axis of an xy plane. At time t = 0, the center is above the
origin, and a pink spot P which rotates with the wheel lies b units below the center
of the wheel. Show that if an x', y' coordinate system, with origin at 0, travels
with the wheel but keeps its unit vectors i' and j' in the directions of i and j,
then
O'P = -b sin wti - b cos wtj.

Show that, because the wheel rolls without slipping,

00' = awti + aj.

Show that the vector r running from 0 to P is

r = (awt - b sin wt)i + (a - b cos wt)j.

Show that

v = (aw - bw cos wt)i + (bw sin wt)j

a = bw2(sin wti + cos wtj).