Question

Which polar coordinates represent the same point as the rectangular coordinate left parenthesis 1 comma negative 3 right parenthesis? Select all that apply.
(1 point)
Responses

left parenthesis start root 10 end root comma 71.6 degrees right parenthesis
Image with alt text: left parenthesis start root 10 end root comma 71.6 degrees right parenthesis

left parenthesis start root 10 end root comma 288.4 degrees right parenthesis
Image with alt text: left parenthesis start root 10 end root comma 288.4 degrees right parenthesis

left parenthesis negative start root 10 end root comma 116.6 degrees right parenthesis
Image with alt text: left parenthesis negative start root 10 end root comma 116.6 degrees right parenthesis

left parenthesis negative start root 10 end root comma 108.4 degrees right parenthesis

Which polar equation represents an ellipse?
(1 point)
Responses

lower r equals start fraction 3 over 2 minus 5 sine theta end fraction
Image with alt text: lower r equals start fraction 3 over 2 minus 5 sine theta end fraction

lower r equals start fraction 2 over 2 minus sine theta end fraction
Image with alt text: lower r equals start fraction 2 over 2 minus sine theta end fraction

lower r equals start fraction 4 over 3 plus 3 cosine theta end fraction
Image with alt text: lower r equals start fraction 4 over 3 plus 3 cosine theta end fraction

lower r equals start fraction 5 over 3 plus 4 cosine theta end fraction
Image with alt text: lower r equals start fraction 5 over 3 plus 4 cosine theta end fraction

Answer 1

To convert the rectangular coordinate \((1, -3)\) to polar coordinates, we first calculate the radius \(r\) and the angle \(\theta\).

1. Calculate the radius \(r\):
\[
r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

2. Calculate the angle \(\theta\):
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-3}{1}\right) = \tan^{-1}(-3)
\]
This angle is in the fourth quadrant since \(x\) is positive and \(y\) is negative. We can find the angle in degrees using a calculator:
\[
\theta \approx -71.6^\circ
\]
To convert this to positive angle, we can add \(360^\circ\):
\[
\theta = 360^\circ - 71.6^\circ = 288.4^\circ
\]

Thus, the polar coordinates \((\sqrt{10}, 288.4^\circ)\) represent the same point. Additionally, there could be an equivalent representation in polar form using a negative radius and adding \(180^\circ\) to the angle:
\[
\theta = -71.6^\circ + 180^\circ = 108.4^\circ
\]

So the valid polar coordinate representations that correspond to the point \((1, -3)\) are:
- \((\sqrt{10}, 288.4^\circ)\)
- \((- \sqrt{10}, 108.4^\circ)\)

### Responses for the first question:
- \((\sqrt{10}, 288.4^\circ)\)
- \((- \sqrt{10}, 108.4^\circ)\)


### For the second question about polar equations representing an ellipse:
A polar equation of the form:
\[
r = \frac{ed}{1 + e \cos(\theta)} \quad \text{or} \quad r = \frac{ed}{1 + e \sin(\theta)}
\]
where \(e < 1\) represents an ellipse.

Let’s check each option for the condition \(e < 1\):

1. **\(r = \frac{3}{2 - 5 \sin(\theta)}\)** \( \rightarrow\) Here, \(e = 5\) \(\rightarrow\) Not an ellipse (since \(e \geq 1\)).

2. **\(r = \frac{2}{2 - \sin(\theta)}\)** \(\rightarrow\) Here, \(e = 1\) \(\rightarrow\) Not an ellipse (since \(e \geq 1\)).

3. **\(r = \frac{4}{3 + 3 \cos(\theta)}\)** \(\rightarrow\) Here, \(e = 1\) \(\rightarrow\) Not an ellipse (since \(e \geq 1\)).

4. **\(r = \frac{5}{3 + 4 \cos(\theta)}\)** \(\rightarrow\) Here, \(e = \frac{4}{5} < 1\) \(\rightarrow\) This represents an ellipse.

### Responses for the second question:
- \(r = \frac{5}{3 + 4 \cos(\theta)}\)