To analyze the graph of \( r = -14 \cos \theta \), we can convert this polar equation into Cartesian coordinates to determine the characteristics.
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Understanding the Equation:
The equation \( r = -14 \cos \theta \) suggests that the polar graph is influenced by the cosine function, and the negative sign indicates a reflection over the origin. -
Identifying Features:
Polar equations of the form \( r = a \cos \theta \) represent circles when \( a \) is positive. The general form \( r = a \cos \theta \) corresponds to a circle centered at \( (a/2, 0) \) with a radius of \( |a|/2 \). -
Radius and Center Calculations:
Here, \( a = -14 \). Referring to the form, we can extract:- The center can be identified at \( (-7, 0) \) because: \[ \text{Center} = \left(-\frac{a}{2}, 0\right) = \left(-\frac{-14}{2}, 0\right) = (-7, 0) \]
- The radius is \( |a|/2 \): \[ \text{Radius} = \frac{|-14|}{2} = 7 \]
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Analyzing the Negative Radius:
The negative radius implies a reflection over the origin. Thus, although the center is at \( (-7, 0) \), the graph ultimately still describes a circle.
Now, combining these characteristics:
- The center is at \( (-7, 0) \).
- The radius is \( 7 \), not \( 14 \).
Thus, none of the options perfectly match all the derived characteristics, especially in radius. The correct interpretation, if the options had included the radius as \( 7 \) rather than \( 14 \), would have been:
- Circle, radius of 7, center at \( (-7, 0) \).
However, based on the closest option possible based on description:
- If asked which is more accurate from the given options, we would have to state that none completely fit but recognize that it’s around the \( (-7, 0) \) center with the properties of a circle.
If one must choose among the closest options, the one that accurately describes the center would be: Circle, radius of 14, center at \( (-7, 0) \) would be the misinterpreted statement that tries to identify the properties rounded to a certain figure.
Make sure to clarify any inconsistencies in the problem options next!