Answer
(a) e = 5/4
(b) Hyperbola
(c) y = 8/5
(d) Image below:
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Work Step by Step
(a) $$r = \frac {8}{4 + 5sin(\theta)} \div \frac{4}{4} = \frac {2}{1
+ \frac 54 sin(\theta)}$$ Based on the original equation: $$\frac{ed}{1 + esin(\theta)}$$ $$e = 5/4$$ (b) $e = 5/4 $, therefore, this equation represents a hyperbola.
(c) The equation has a $+sib (\theta)$ and the focus is at (0,0), thus, the equation of the directrix must be in the "$y = +c$" pattern. In this case: $c = d$: $$ed = 2 \longrightarrow d = \frac{2}{e} = \frac{2}{5/4} = 8/5 $$ $$y = 8/5$$
(d) Plot the points where $\theta = 0$, $\theta = \pi/2$, $\theta = 3\pi/2$
$$r = \frac {8}{4 + 5sin(0)} \longrightarrow (2, 0)$$ $$r = \frac {8}{4 + 5sin(\pi/2)} \longrightarrow (8/9, \pi/2)$$ $$r = \frac {8}{4 + 5sin(\pi)} \longrightarrow (2, \pi)$$ $$r = \frac {8}{4 + 5sin(3\pi/2)} \longrightarrow (-8, 3\pi/2)$$
Draw a hyperbola that passes through these points.
