The problem asks us to identify the curve represented by the polar equation $r = \frac{-4}{\cos \theta}$, state its eccentricity if it is a conic, and sketch the graph.

1. Problem Description

The problem asks us to identify the curve represented by the polar equation $r = \frac{-4}{\cos \theta}$, state its eccentricity if it is a conic, and sketch the graph.

2. Solution Steps

First, we rewrite the equation as $r \cos \theta = -4$.

Recall the relationship between polar and rectangular coordinates:

$x = r \cos \theta$

$y = r \sin \theta$

Substituting $x$ for $r \cos \theta$, we have:

$x = -4$

This represents a vertical line. Since a line is a degenerate conic with eccentricity $e=1$.

3. Final Answer

Curve: Vertical line

Eccentricity: $e=1$

Equation: $x=-4$