Calculus-Sec-10-3-Solution

10.3    Polar Coordinates                           by SGLee - HSKim - JHLee

1-4. Plot the point whose polar coordinates are given. Then, find the Cartesian coordinates of the point.

1.  =

2.  =

3.  =

.

4.  =

.

5-8. The Cartesian coordinates are given.

Find two other pairs of polar coordinates of the point, one with   and other .

5.

, and .

Since  is in the fourth quadrant, the polar coordinates are

⑴  : , ⑵  : .

6.

, and .

Since  is in the second quadrant, the polar coordinates are

⑴  : , ⑵  : .

7.

, and .

Since  is in the fourth quadrant, the polar coordinates are

⑴  : , ⑵  : .

8.

, and .

Since  is in the second quadrant, the polar coordinates are

⑴  : , ⑵  : .

9. Find a formula for the distance between the points with polar coordinates  and .

Let

Let  be the e distance between  and .

10-11. Find a polar equation for the curve represented by the given Cartesian equation.

10.

.

11.

.

12.

.

13-14. Find a Cartesian equation for the curve represented by the given polar equation.

13.

,

a circle of radius  centered at .

The first two equations are actually equivalent since

.

But  gives the point  when .

Thus, the single equation  is equivalent to the compound condition ().

14.

.

15.

16-25. Sketch the curve with the given polar equation.

16.

24.

25.

26. Show that the polar curve  (called a conchoid) has the line

as a vertical asymptote by showing that . Use this fact to sketch the conchoid.

.

Now, ,

so .

Also, , so

.

Therefore,  is a vertical asymptote.

27. Show that the curve  (also a conchoid) has the line

as a horizontal asymptote by showing that . Use this fact to help sketch conchoid.

.

Now, ,

so .

Also, ,

so .

Therefore,  is a horizontal asymptote.

28. Show that the curve  (called a cissoid of Diocles) has the line

as a vertical asymptote. Show also that the curve lies entirely within the vertical strip .

Use these facts to sketch the cissoid.

29-32. Find the slope of the tangent line at the given point.

29.

.

30.

.

31.

.

32.

33-36. Find the points on the given curve where the tangent line is horizontal or vertical.

33.

.

.

So the tangent is vertical at  and .

. So the tangent is horizontal at  and .

34.

35.

36.

37. Show that the polar equation  represents a circle and find its center and radius.

.

Then the polar equation  represents a circle and its center : , radius :.