Find all values of $c$such that the graphs of $r=c$and $r=-c$are the same in a polar coordinate system.

The equations $r=c$and $r=-c$.

Consider the equations $r=c$and $r=-c$.

The objective is to show that $r=c$and $r=-c$are same in polar coordinate system.

For any real number $c$the polar equation $r=c$describes the set of points $\left|c\right|$units from the pole.

Thus $r=c$is a circle with radius $\left|c\right|centered$at the pole.

Now for any real number $-c$the polar equation $r=-c$describes the set of points $\left|c\right|$units from the pole.

Thus $r=-c$is a circle with radius $\left|c\right|$centered at the pole.

Here localid="1652686412744" $\mathit{r}$represents the length of the initial ray, that means the length of the initial ray is $c$units if $r=c$or $r=-c$since the length is positive.

Example:

If $r=1$or $r=-1$

The length of the initial ray is $|-1|$, a circular image created by the fact that our equation does not give us a restriction on the size of theta or angle. So it becomes a circle with the radius 1.

That means the equations $r=c$and $r=-c$are same in polar coordinate system.

Therefore $r=c$and $r=-c$are same in polar coordinate system.

Hence the explanation.