Final answer:
The equation of the hyperbola with directrices at x=±2 and foci at (5, 0) and (−5, 0) is x²/21 - y²/46 = 1 (option C).
Step-by-step explanation:
The equation of a hyperbola with directrices at x = ±2 and foci at (5, 0) and (-5, 0) can be found using the formula for the standard form of a hyperbola. The standard form equation of a hyperbola with horizontal transverse axis is given by:
(x-h)²/a² - (y-k)²/b² = 1
where (h, k) are the coordinates of the center, and a and b determine the shape and size of the hyperbola.
In this case, the center is (0,0). The distance between the center and each focus is the value of 'c', which is 5. The distance between the center and each directrix is the value of 'd', which is 2. The value of 'a' can be found using the equation a = √(c² - d²), and the value of 'b' can be found using b = √(a² + c²).
Substituting the values into the standard form equation will give us the equation of the hyperbola.
Calculating 'a': a = √(c² - d²) = √(5² - 2²) = √(25 - 4) = √21
Calculating 'b': b = √(a² + c²) = √(21 + 5²) = √(21 + 25) = √46
The equation of the hyperbola is therefore x²/21 - y²/46 = 1 (option C).