(Solved): According to Kepler's first law, a comet should have an elliptic, parabolic, or hyperbolic orbit (w ...

According to Kepler's first law, a comet should have an elliptic, parabolic, or hyperbolic orbit (with gravitational attractions from the planets ignored). In suitable polar coordinates, the position \( (r, 9) \) of a comet satisfies an equation of the form \( r=\beta+e(r \cdot \cos \theta) \), where \( \beta \) is a constant and \( \theta \) is the eccentricity of the orbit, with \( 0 \leq \theta<1 \) for an ellipse, \( \theta=1 \) for a parabola, and e \( >1 \) for a hyperbola. Suppose observations of a newly discovered comet provide the data below. Determine the type of orbit, and predict where the comet will be when \( \theta=4.1 \) (radians). The comet has orbit. When \( 2=4.1 \) (radians), the comet will be at \( r= \) (Round to two decimal places as needed.)