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
