Question

Find the vertices and foci of the hyperbola with the equation ( fracx²/16 - fracy²/48 = 1 ).

a) Vertices: (0, ± 4); Foci: (0, ±8)
b) Vertices: (0, ± 8); Foci: (0, ± 4)
c) Vertices: (± 4, 0); Foci: (± 8, 0)
d) Vertices: (± 8, 0); Foci: (± 4, 0)

Answer 1

The vertices of the hyperbola are (±8, 0) and the foci are (±4√2, 0).

Step-by-step explanation:

The equation of the hyperbola is in the form x²/16 - y²/48 = 1. By comparing this equation to the standard form of a hyperbola, we can determine that the hyperbola has a horizontal transverse axis and the center at the origin (0,0).

The vertices of the hyperbola can be found by calculating the distance from the center to the vertices using the formula a = √(b² + c²), where a is the distance from the center to the vertices and b and c are the lengths of the conjugate and transverse axes, respectively. In this case, a = √(16 + 48) = √64 = 8. So, the vertices are (±8, 0).

Similarly, the distance from the center to the foci can be found using the formula c = √(a² + b²). In this case, c = √(48 - 16) = √32 = 4√2. So, the foci are (±4√2, 0).