To find the equation of the hyperbola, we start with the provided information about the distance between the foci and the eccentricity. We know the following properties of hyperbolas:
The standard form of the equation of a hyperbola with its major axis along the x-axis is:
‌−‌=1Where:
a is the distance from the center to each vertex on the x-axis.
b is associated with the distances in the
y-direction.
The distance between the foci is
2c.
c2=a2+b2 (relationship between
a,b, and
c for hyperbolas).
The eccentricity
e is given by
e=‌.
Given that the distance between the foci
(2c) is 16 :
2c=16⟹c=8And the eccentricity
(e) is
√2 :
e=‌=√2⟹a=‌=‌=4√2Now, using
c2=a2+b2, we find
b2 :
‌c2=a2+b2‌64=(4√2)2+b2‌64=32+b2‌b2=32Thus, the equation of the hyperbola becomes:
‌‌−‌=1‌‌−‌=1‌‌−‌=1‌x2−y2=32 This corresponds to Option A:
x2−y2=32The correct answer is Option
A:x2−y2=32.