Determine two pairs of polar coordinates for the point (5, 5) with 0° ≤ θ < 360°.

a. (5 square root 2, 225°), (-5 square root 2, 45°)

b. (5 square root 2, 315°), (-5 square root 2, 135°)

c. (5 square root 2, 135°), (-5 square root 2, 315°)

d. (5 square root 2, 45°), (-5 square root 2, 225°)

find an equation in standard form for the hyperbola with vertices at (0, ±9) and foci at (0, ±10).

a. y squared over 81 minus x squared over 100 = 1
b. y squared over 81 minus x squared over 19 = 1
c. y squared over 19 minus x squared over 81 = 1
d. y squared over 100 minus x squared over 81 = 1

First part

The answer is (5 square root 2, 45°), (-5 square root 2, 225°) ⇒ answer (d)

Second part

The equation in standard form for the hyperbola is y²/81 - x²/19 = 1 ⇒ answer(b)

Step-by-step explanation:

First part:

* Lets study the Polar form and the Cartesian form

- The important difference between Cartesian coordinates and

  polar coordinates:

# In Cartesian coordinates there is exactly one set of coordinates

  for any given point.

# In polar coordinates there is an infinite number of coordinates

   for a given point. For instance, the following four points are all

   coordinates for the same point.

# In the polar the coordinates the origin is called the pole, and

  the x axis is called the polar axis.

# The angle measurement θ can be expressed in radians

   or degrees.

- To convert from Cartesian Coordinates (x , y) to

  Polar Coordinates (r , θ)

# r = ± √(x² + y²)

# θ = tan^-1 (y / x)

* Lets solve the problem

- The point in the Cartesian coordinates is (5 , 5)

∵ x = 5 and y = 5

∴ r = ± √(5² + 5²) = ± √50 = ± 5√2

∴ tanФ = (5/5) = 1

∵ tanФ is positive

∴ Angle Ф could be in the first or third quadrant

∵ Ф = tan^-1 (1) = 45°

∴ Ф in the first quadrant is 45°

∴ Ф in the third quadrant is 180 + 45 = 225°

* The answer is (5√2 , 45°) , (-5√2 , 225°)

Second part:

* Lets study the standard form of the hyperbola equation

- The standard form of the equation of a hyperbola with  

  center (0 , 0) and transverse axis parallel to the y-axis is

  y²/a² - x²/b² = 1, where

•the length of the transverse axis is 2a

•the coordinates of the vertices are (0 , ±a)

•the length of the conjugate axis is 2b

•the coordinates of the co-vertices are (±b , 0)

•      the coordinates of the foci are (0 , ± c),  

•the distance between the foci is 2c, where c² = a² + b²

* Lets solve our problem

∵ The vertices are (0 , 9) and (0 , -9)

∴ a = ± 9 ⇒ a² = 81

∵ The foci at (0 , 10) , (0 , -10)

∴ c = ± 10

∵ c² = a² + b²

∴ (10)² = (9)² + b² ⇒ 100 = 81 + b² ⇒ subtract 81 from both sides

∴ b² = 19

∵ The equation is  y²/a² - x²/b² = 1

∴  y²/81 - x²/19 = 1

* The equation in standard form for the hyperbola is y²/81 - x²/19 = 1

There is not a fair opportunity.

Step-by-step explanation: this is because when you look at the table, it was 65, then 70 people that did not get the art class they wanted. But then it says that 135 people did not get the art class they wanted. That doesn’t seem fair because there is a huge difference between 65 and 135, almost twice as much.