Question

Which is the equation of a hyperbola with directrices at y = ± 2 and foci at (0, 6) and (0, 6)?

a. ( y²/4 - x²/36 = 1 )
b. ( y²/4 - x²/36 = -1 )
c. ( x²/4 - y²/36 = 1 )
d. ( x²/4 - y²/36 = -1 )

Answer 1

The equation of the hyperbola with directrices at y = ±2 and foci at (0, 6) and (0, 6) is c) x²/4 - y²/36 = 1.

Step-by-step explanation:

The equation of a hyperbola with directrices at y = ±2 and foci at (0, 6) and (0, 6) is x²/4 - y²/36 = 1 (option c).

A hyperbola is defined by the equation (x - h)²/a² - (y - k)²/b² = 1 (for a horizontal hyperbola) or (y - k)²/a² - (x - h)²/b² = 1 (for a vertical hyperbola). In this case, the center of the hyperbola is at (h, k) = (0, 0), the distance from the center to the foci is c = 6, and the distance from the center to the vertices is a = 2.

Using the formula c² = a² + b², we can find that b² = c² - a² = 36 - 4 = 32. Substituting these values into the equation of a hyperbola, it becomes x²/4 - y²/36 = 1.