32.6k views
0 votes
Describe the graph represented by the equation r = 4 / 2 + 3 sin theta

Describe the graph represented by the equation r = 4 / 2 + 3 sin theta-example-1

1 Answer

7 votes

Answer:

Hyperbola, horizontal directricx at a distance 4/3 units above the pole

Explanation:

* The polar equation for a conic i

- For a conic with a focus at the origin, if the directrix is

x = ± p, where p is a positive real number, and the eccentricity

is a positive real number e, the conic has a polar equation

# r = ep/(1 ± e cosФ)

- For a conic with a focus at the origin, if the directrix is

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

is a positive real number e, the conic has a polar equation

# r = ep/(1 ± e sinФ)

- For a conic with eccentricity e,

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

# if e = 1, the conic is a parabola

# if e > 1, the conic is an hyperbola

* Lets solve the problem

∵ 4 = r/(2 + 3sinФ)

- From the rule above

∴ Directrix is y = ± p

- Divide up and down by 2 to make the 1st term in the

bracket down = 1

∴ r = 2/(1 + (3/2)sinФ)

- Compare it with the rule

∴ e = 3/2 > 1

∴ The conic is hyperbola

∵ ep = 2

∴ p = 2 ÷ e = 2 ÷ 3/2 = 2 × 2/3 = 4/3

∴ Directrix is y = ± 4/3

* Now we can describe the graph

- Hyperbola, horizontal directricx at a distance 4/3 units above the pole

User Marketer
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories