Answer:
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
of the apartment per hour. Similarly Liz cleans 
of the appartment per hour. So together then can clean 
of the apartment for hour. Therefor it will take them 
hours to clean.
can be used as a check if needed.