Answer:
The greatest perimeter is in (D) ⇒ the perimeter = 60 units
Step-by-step explanation:
* In the hyperbola
- The standard form of the equation of a hyperbola with
center (h , k) and transverse axis parallel to the x-axis is
(x - h)²/a² - (y - k)²/b² = 1
∵ the length of the transverse axis is 2a
∵ the length of the conjugate axis is 2b
∴ The perimeter of asymptote rectangle is 2(2a + 2b)
* Lets check the answers to find the greatest perimeter
A) (x - 4)²/11² - (y + 2)²/3² = 1
* Compare it with the standard form equation
∵ a = 11 ⇒ 2a = 22
∵ b = 3 ⇒ 2b = 6
∴ The perimeter = 2(22 + 6) = 56
B) (x - 2)²/4² - (y + 1)²/10² = 1
* Compare it with the standard form equation
∵ a = 4 ⇒ 2a = 8
∵ b = 10 ⇒ 2b = 20
∴ The perimeter = 2(8 + 20) = 56
C) (x + 5)²/5² - (y - 3)²/9² = 1
* Compare it with the standard form equation
∵ a = 5 ⇒ 2a = 10
∵ b = 9 ⇒ 2b = 18
∴ The perimeter = 2(10 + 18) = 56
D) (x - 7)²/8² - (y - 2)²/7² = 1
* Compare it with the standard form equation
∵ a = 8 ⇒ 2a = 16
∵ b = 7 ⇒ 2b = 14
∴ The perimeter = 2(16 + 14) = 60
* The greatest perimeter is in (D)
