Solve the problem.
-Given that the polar equation $\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }$ models the orbits of the planets about the sun, estimate the closest approach of a planet for which $\mathrm { a } = 39$ and $\mathrm { e } = 0.78$ to the sun.
A) 9 astronomical units
B) 15 astronomical units
C) 4 astronomical units
D) 69 astronomical units

Question

Quizplus · Accepted Answer

The closest approach, or perihelion, of a planet to the sun in its orbit is found when $\theta = 0$, which minimizes the denominator in the given polar equation. Substituting $\theta = 0$, $a = 39$, and $e = 0.78$ into the equation gives $r = \frac{39(1-0.78^2)}{1+0.78\cos(0)}$, which simplifies to approximately 9 astronomical units.

Solve the Problem r=a(1−e2)1+ecos⁡θ\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }r=1+ecosθa(1−e2)​

Solve the Problem $\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }$