Q. 7 - Calculus

Question

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

Step-by-Step Solution

Answer

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.

Step 1: Given information

The equations $r=c$ and $r=-c$.

Step 2: Simplification

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 $\left|c\right| c e n t e r e d$ 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 $\mathit{r}$ 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.