Which hyperbola’s asymptote rectangle has the greatest perimeter?

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)

step-by-step explanation: