Answer
When $\theta$ is close to $0^{\circ}$, we can see that Pluto's distance from the sun is slightly closer than Neptune's distance from the sun. Therefore, Pluto is not always farthest from the sun.
Work Step by Step
$r = \frac{a~(1-e^2)}{1+e~cos~\theta}$
Neptune:
$a = 30.10$
$e = 0.009$
$\theta = 0^{\circ}~~~~~~~r = 29.83$
$\theta = 90^{\circ}~~~~~~r = 30.10$
$\theta = 180^{\circ}~~~~~r = 30.37$
$\theta = 270^{\circ}~~~~~r = 30.10$
Pluto:
$a = 39.40$
$e = 0.249$
$\theta = 0^{\circ}~~~~~~~r = 29.59$
$\theta = 90^{\circ}~~~~~~r = 36.96$
$\theta = 180^{\circ}~~~~~r = 49.21$
$\theta = 270^{\circ}~~~~~r = 36.96$
We can see the graph of the two orbits below:
When $\theta$ is close to $0^{\circ}$, we can see that Pluto's distance from the sun is slightly closer than Neptune's distance from the sun. Therefore, Pluto is not always farthest from the sun.
