Question

select the representations that do not change the location of the point (9,140°).
choose the correct answer below. choose all that apply.
a. (-9,320°)
b. (-9,220°)
c. (-9,-40°)
d. (9,-220°)

Explanation:

Step1: Check r and angle relation

For polar coordinates $(r, \theta)$, equivalent coordinates satisfy:

1. $(r, \theta + 360^\circ k)$ or $(-r, \theta + (2k+1)180^\circ)$ for integer $k$.

Step2: Analyze Option A

Original: $(9,140^\circ)$. Option A: $(9,320^\circ)$. $320^\circ -140^\circ=180^\circ
eq360^\circ k$, so not equivalent.

Step3: Analyze Option B

Option B: $(-9,220^\circ)$. $-r=-9$, $\theta +180^\circ=140^\circ+180^\circ=320^\circ
eq220^\circ$, so not equivalent.

Step4: Analyze Option C

Option C: $(-9,-40^\circ)$. $-r=-9$, $\theta +180^\circ=320^\circ$. $-40^\circ +360^\circ=320^\circ$, so $(-9,-40^\circ)=(-9,320^\circ)$, which is equivalent.

Step5: Analyze Option D

Option D: $(9,-220^\circ)$. $-220^\circ +360^\circ=140^\circ$, so $(9,-220^\circ)=(9,140^\circ)$, equivalent.

Answer:

A. $(-9,320^\circ)$, B. $(-9,220^\circ)$