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 t + 2 for − ≤ t ≤ −1
3

xt ( ) = +− 1 tan  π ( − t )  for −< t < 0
2

1
1
1
 3  3 
 1  π 2 
t
1
1
0
 −+ tan ( − t )  for << 1
 3  3
0  for −≤ t ≤ −1
3
 1  π 
yt ( ) =  tan ( − t ) for −<1 t < 0
2
1
 3  3 
 1  π π 2 
t
 tan  1 ( − t )  for 0 << 1
 3 3
Homework Problems
The following set of problems provides the mathematical basis for under-
standing the graphical display on the screen of an oscilloscope, when in the
x-y mode.
Pb. 1.7 To put the quadratic expression
2
Ax + Bxy + Cy + Dx + Ey + F = 0
2
in standard form (i.e., to eliminate the x-y mixed term), make the transformation
x = x′cos( )θ − y′sin( )θ

y = x′sin( )θ + y′cos( )θ

Show that the mixed term is eliminated if cot(2θ≡ ( −AC ) .
)
B
Pb. 1.8 Consider the parametric equations

C: x = a cos(t), y = b sin(t + ϕ), and 0 < t < 2π

where the initial point is at x = a, y = b sin(ϕ), and the final point is at x = a,
y = b sin(ϕ).
a. Obtain the equation of the curve in the form f(x, y) = 0.
b. Using the results of Pb. 1.7, prove that the ellipse inclination angle
is given by:

cot(2θ ≡ (a 2 − b 2 )
)
2ab sin( ) ϕ

© 2001 by CRC Press LLC