Find the polar coordinates of the point(s) of intersection of the given curves for 0 $$0 \leq \theta

Question

Find the polar coordinates of the point(s) of intersection of the given curves for 0 $$0 \leq \theta < 2 \pi$$
-$$r = \frac { a \left( 1 - e ^ { 2 } \right) } { 1 + e \cos \theta }$$ Given that the polar equation r models the orbits of the planets, estimate the closest approach of Pluto's orbit, for which a = 39.4 and e = 0.249, to the sun.
A) About 39 astronomical units
B) About 74 astronomical units
C) About 30 astronomical units
D) About 50 astronomical units

Quizplus · Accepted Answer

The closest approach, or perihelion, of an orbit in polar coordinates given by $r = \frac { a \left( 1 - e ^ { 2 } \right) } { 1 + e \cos \theta }$ occurs when $\cos \theta = 1$, which corresponds to $\theta = 0$. Substituting $a = 39.4$ and $e = 0.249$ into the equation and evaluating at $\cos \theta = 1$ gives the closest approach distance. The calculation is $r = \frac { 39.4 \left( 1 - 0.249 ^ { 2 } \right) } { 1 + 0.249 }$, which approximates to about 30 astronomical units.

Find the Polar Coordinates of the Point(s) of Intersection of the Given