Answer
(a) e = 4/5
(b) Ellipse
(c) y = -1
(d) Image below:
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Work Step by Step
(a)
$$r = \frac {4}{5 - 4sin(\theta)} \div \frac{5}{5} = \frac {4/5}{1 - \frac 45 sin(\theta)}$$
Based on the original equation:
$$\frac{ed}{1 - esin(\theta)}$$
$$e = 4/5$$
(b)
$e < 1 $, therefore, this equation represents an ellipse.
(c)
The equation has a $-sin(\theta)$ and the focus is at (0,0), thus, the equation of the directrix must be in the "$y = -c$" pattern.
In this case: $c = d$:
$$ed = \frac 45 \longrightarrow d = \frac{4/5}{e} = \frac{4/5}{4/5} = 1$$
$$y = -1$$
(d)
Plot the points where $\theta = 0$, $\theta = \pi/2$, $\theta = \pi$ and $\theta = 3\pi/2$
$$r = \frac{4}{5 - 4sin(0)} = 4/5 \longrightarrow (\frac 45, 0)$$
$$r = \frac{4}{5 - 4sin(\pi/2)} = 4 \longrightarrow ( 4, \pi/2)$$
$$r = \frac{4}{5 - 4sin(\pi)} = 4/5 \longrightarrow (\frac 45, \pi)$$
$$r = \frac{4}{5 - 4sin(3\pi/2)} = 4/9 \longrightarrow ( 4/9, 3\pi/2)$$
Draw an ellipse that pass through these points.
