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

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

Step-by-step 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

good job you smart son of a gun you are smarter that einstein. 7

circle area = pi * radius^2

if diameter = 10 then radius = 5

circle area = pi * 5^2

circle area = pi * 25

circle area = 78.5398163397 square units

step-by-step explanation: