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 smallest possible distance between Earth, for which a = 1.00 and e = 0.017, and Venus, for which a = 0.78 and
E = 0.007.
A) About 0.2 astronomical units
B) Zero astronomical units
C) About 2 astronomical units
D) About 0.4 astronomical units

Quizplus · Accepted Answer

The smallest distance between Earth and Venus can be estimated by finding the minimum value of the difference in their respective distances from the Sun at any point in their orbits. Given the parameters for Earth (a = 1.00, e = 0.017) and Venus (a = 0.78, e = 0.007), and considering the formula for the orbits, the smallest distance occurs when both planets are at their perihelion (closest point to the Sun) on the same line from the Sun. The difference in their perihelion distances (calculated using $r = \frac { a \left( 1 - e ^ { 2 } \right) } { 1 + e \cos \theta }$ with $\theta = 0$ for perihelion) gives an estimate of about 0.2 astronomical units, considering the slight eccentricities of their orbits.

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