**Polar Coordinates**

**Sol A** gives the polar equation , so taking square
roots gives . Therefore is a polar equation for the circle
(since and represent the same circle).

**Sol B** Multiplying both sides of by gives
, so . To put this equation in standard form, we
can subtract from both sides to get and then complete the
square to obtain
or .

**Sol C** Taking gives , so is one set of polar
coordinates for the point. Taking gives , so is
another set of polar coordinates for the point.

(More generally, we can take and to be any odd multiple of , or and to be any even multiple of .

**Sol 1**
, so we can take .
Since
and is in
Quadrant IV, we can take
. Therefore
is a set of polar coordinates for the point.

**Sol 2**
, so we can take
. Since
and
is in Quadrant III, we can take .
Therefore
is a set of polar coordinates for the point.

**Sol 3**
and
, so the point has
rectangular coordinates
.

**Sol 4** gives , so
and
therefore .

**Sol 5** gives , so
and
therefore .

**Sol 6** gives
or , so
and therefore .

**Sol 7** Multiplying both sides of
by gives
or
. Then
or , so squaring both sides gives
, and
thus
.

**Sol 8** Setting the two expressions for equal to each other gives
, so and
.
Therefore we can take or , and then the
corresponding value of is given by
.
Therefore the curves intersect at the points with polar coordinates
and , and they also intersect at the origin
by inspection.

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