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: ‌
x2
a2
−‌
y2
b2
=1 Where: 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=‌
c
a
. Given that the distance between the foci (2c) is 16 : 2c=16⟹c=8 And the eccentricity (e) is √2 : e=‌
c
a
=√2⟹a=‌
c
√2
=‌
8
√2
=4√2 Now, using c2=a2+b2, we find b2 : ‌c2=a2+b2 ‌64=(4√2)2+b2 ‌64=32+b2 ‌b2=32 Thus, the equation of the hyperbola becomes: ‌‌
x2
a2
−‌
y2
b2
=1 ‌‌
x2
(4√2)2
−‌
y2
32
=1 ‌‌
x2
32
−‌
y2
32
=1 ‌x2−y2=32 This corresponds to Option A: x2−y2=32 The correct answer is Option A:x2−y2=32.