Describe the graph of `r=(3)/(cos theta)`

Options

A. a parabola
B. an ellipse
C. a circle
D. a vertical line

Explanation

To describe the graph of the polar equation \( r = \frac{3}{\cos \theta} \), we will convert this equation into rectangular coordinates and analyze its implications step by step.

### Step 1: Start with the polar equation
We have the polar equation:
\[ r = \frac{3}{\cos \theta} \]

### Step 2: Use the relationship between polar and rectangular coordinates
We know that in polar coordinates:
\[ x = r \cos \theta \]
Substituting the expression for \( r \) into this equation gives:
\[ x = \left(\frac{3}{\cos \theta}\right) \cos \theta \]

### Step 3: Simplify the equation
Now simplify the expression:
\[ x = 3 \]

### Step 4: Interpret the result
The equation \( x = 3 \) represents a vertical line in the rectangular coordinate system. This line is located at \( x = 3 \) and extends infinitely in the positive and negative \( y \)-directions.

### Step 5: Identify holes in the graph
Next, we need to consider the values of \( \theta \) where \( \cos \theta = 0 \). This occurs at:
\[ \theta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
At these angles, \( r \) becomes undefined (since division by zero is not allowed), which means there are holes in the graph at these points.

### Conclusion
The graph of the equation \( r = \frac{3}{\cos \theta} \) is a vertical line at \( x = 3 \) with holes at points where \( \theta = \frac{\pi}{2} + n\pi \).

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