Question

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

A. x^2/12−y^2​/24=1
B. x^2​/12−y^2​/48=1
C. y^2​/12−x^2​/24=1
D. y^2​/48−x^2​/12=1

Answer 1

The equation of the hyperbola is y²/36 - x²/4 = 1.

Step-by-step explanation:

The equation of a hyperbola with directrices at y = ±2 and foci at (0, 6) and (0, −6) can be determined using the standard form of the equation for a hyperbola. The standard form for a hyperbola with the center at the origin is given by:

y²/a² - x²/b² = 1

where a represents the distance from the center to the vertices and b represents the distance from the center to the asymptotes. In this case, because the directrices are at y = ±2 and the foci are at (0, 6) and (0, -6), we know that a = 6 and b = 2. Therefore, the equation of the hyperbola is:

y²/36 - x²/4 = 1