Which conic section is represented by the polar equation r=3 / (4+2sintheta) ?

A. ellipse

B. hyperbola

C. parabola

D. circle

The conic is an ellipse ⇒ answer A

Step-by-step explanation:

* Lets explain how to solve the problem

- The polar form equation of a conic with a focus at the origin, the

directrix is y = ± p where p is a positive real number, and the

eccentricity is a positive real number e is r = ep / (1 ± e sin Ф)

# If 0 ≤ e <1, then the conic is an ellipse

# If e = 1, then the conic is a parabola

# If e > 1, then the conic is an hyperbola

- Lets solve the problem

∵ The equation of the conic is r = 3 / (4 + 2 sin Ф)

∵ The form of the equation is r = ep / (1 ± e sin Ф)

- We must divide up and down by 4 to make the 1st term of the

denominator equal 1

∴ r = (3/4) / (1 + (2/4) sin Ф)

∴ ep = 3/4

∴ e = 2/4

- Lets use the rules above to identify the type of the conic

∵ e = 2/4 = 1/2

∴ 0 ≤ e < 1

∴ The conic is an ellipse