Explanation

(b)
Give equation is $\frac{1}{r} = \frac{1}{8} + \frac{3}{8}\cos{\theta,}$ it can be rewriter as $\frac{8}{r} = 1 + 3\cos\theta$, which is the form of $\frac{1}{r} = 1 + e\cos\theta$
On comparing, we get
$e = 3 > 1$
$\therefore\ \$Given equation represents a hyperbola.