Exercises

1. Transform the point into xy-coordinates.
a.
( 2,p)
b.
( 3,p/6)
b.
( 1,9p)
d.
( -2,2p/3)

2. Transform the point into polar coordinates.
a.
( 2,2)
b.
( -Ö5,-Ö5)
c.
( -Ö3,1)
d.
( -15,0)

Use a polar grid to sketch the graph of the following polar functions. Compare your sketch to the plot produced by the ''polar plotting tool'' in the ''Tools'' section.
3. r = 5,  q = p/4 4.r = -2,  q = p/3
5.
r =  6
p
 q , q = p/6
6.
r =  
q2
p2
, q = p/6
7.
r = 1 + cos(q), q = p/2
8.
r = cos(2q), q = p/4
9.
r = 2+cos(q), q = 2p/3
10.
r = cos(3q), q = 7p/4

Use a polar grid to sketch the graph of the following polar functions. Then find the velocity vector v(q) at the given q. Compare your sketch to the plot produced by the "polar plotting tool" in the "Tools" section.
11.
x2+y2 = 16
12.
x2+3xy = y2
13.
x = 1
14.
y = 1
15.
y = 3x+2
16.
x2-y2 = 1
17.
y = x2
18.
y = x3
19.
x =   y2
4		 -1
20.
y = x2-  1
4
21. x2 + y2 = 2y + 3

22.

x2+y2 = 4x+2y
23.(x-1)2 + y2 = 4

24.

x2+y2 = (x+y)2
25.
xy = 2
26.
3x2+4y2-4x = 4

Identify the eccentricity and the parameter of the following ellipses. Also, find the foci, the semi-major axis, and the semi-minor axis. Then sketch the graph of the ellipse.
27.
r =  12
4-3cos( q)
28.
r =  12
3-cos( q)
29.
r =  2
1-0.2 sin( q)
30.
r =  2
3-2sin( q)
31.
r =  4
2+sin( q)
32.
r =  1
1 + cos( q)
      
33.foci at (0,0) and (6,0)  34.foci at (0,0) and (0,10)
 semi-major axis of 5   semi-major axis of 13

35. Show that a polar function of the form
r =  p
1 - sin( q)

is a parabola.  What is its vertex?

36. Which curve in the xy-plane is represented by the polar equation r = sec(q) tan(q) ?

37. Show that
 d
dq
 eq = -er

38. Show that if r and q are functions of t and r(t) = á r cos(q), r sin(q) ñ , then
 d
dt
 er = eq  dq
dt
   and    d
dt
 eq = -er  dq
dt

39. Show that if r and q are functions of t and r( t) = á rcos( q) ,rsin( q) ñ , then
v
=
 dr
dt
 er  +  r  dq
dt
 eq
a
=
æ
è 		 d2r
dt2
 -r æ
è 		 dq
dt
 ö
ø 		2

 
 ö
ø 		er  +   æ
è 		2  dr
dt
  dq
dt
 +r  d2q
dt2
 ö
ø 		eq
(Hint: See exercise 38).

40. Show that y = mx+b is pulled back into polar coordinates to the function
r =  b
sin( q) -mcos( q)
At what value of q does r not exist? What is significant about this value of q?

41. The orbit of the earth about the sun is an ellipse with the sun's center as one focus. The parameter of the earth's orbit is 185,740,000 miles and the eccentricity is 0.017. What is the semi-major axis, the semi-minor axis, and the foci of the earth's orbit.

42. A satellite has an orbit with a parameter of 4335 miles and an eccentricity of 0.067. If we assume the earth is a sphere with a radius of 3963 miles, what is the closest the satellite comes to the earth?

43. The positions of a planet closest to and farthest from the sun are called its perihelion and aphelion, respectively.

Show that if 0 < e < 1 and if the planet's orbit is given by
r =  p
1-ecos( q)
then the perihelion distance is a( 1-e) and the aphelion distance is a( 1+e) .

44. Pluto is at a distance of 4.43×109 km when it is closest to the Sun and is at a distance 7.37×109 km when it is farthest from the sun. Use the result in exercise 39 to determine the eccentricity of the orbit.

45. Write to Learn: Write a short essay in which you derive and explain the following identities:
a =  r(0) + r(p)
2
 
                
ea =  r(0) - r(p)
2
 
b2 =  r(0) r(p) 
 
                
e =  r(0) - r(p)
r(0) + r(p)
 

46.  The Eccentric Anomaly: The equation in polar coordinates of an ellipse centered on the negative x-axis is
r =  p
1 + e cos( q)

Let a = p/( 1-e2) and let k = p/(1+e) . The auxiliary circle of the ellipse is the circle centered at ( k-a,0) with radius a, and the eccentric anomaly of an ellipse is the angle E formed with by the radius a of the circle that terminates on the vertical line through the point P( r,q) (see figure below):

click image to toggle to an animation

(animation created by Clayton Clark)
The eccentric anomaly is important in celestial mechanics, where we often reparameterize r as a function of the angle E.

  1. Show that r cos( q) = a cos( E) - ae (hint: in the figure above, rcos( p-q) > 0 and acos( E) > 0).

  2. Use (a) and the fact that r + ercos(q) = p to show that
    r = p+ae2-aecos( E)

  3. Show that p = a( 1-e2) , so that (b) implies that
    r = a-aecos( E)

  4. Use the definition of k to show that ae = a-k and that (c) can be written in the form
    r = a-( a-k) cos( E)
    		(4)

  5. Explain why the segment BP in the figure above has a length of
    | BP| = ( a-k) | cos( E)|
    How might (4) and the length of BP be used to suggest a different method for constructing an ellipse?