Solve the problem.
-Find the foci and asymptotes of the following hyperbola: $x ^ { 2 } - y ^ { 2 } = 32$
A) Foci: (4, 0), (-4, 0); Asymptotes: y = x, y = - x
B) Foci: (8, 0), (-8, 0); Asymptotes: y =2x, y = - 2x
C) Foci: (0, 8), (0, -8); Asymptotes: y = x, y = - x
D) Foci: (8, 0), (-8, 0); Asymptotes: y = x, y = - x

Question

Quizplus · Accepted Answer

We can rewrite the equation as $\frac{x^2}{16} - \frac{y^2}{16} = 2$, which is in the standard form $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1$. This tells us that the center of the hyperbola is $(0,0)$, the distance from the center to each vertex is $a=\sqrt{16}=4$, and the distance from the center to each focus is $c=\sqrt{a^2+b^2}=\sqrt{16+16}=4\sqrt{2}$. Therefore, the foci are located at $(\pm4\sqrt{2},0)$.

To find the asymptotes, we note that the slope of the transverse axis is $\pm\frac{b}{a}=\pm1$. Therefore, the two lines passing through the center with slopes $\pm1$ are the asymptotes. These lines have the form $y=\pm x$, so the answer is $D$.

Solve the Problem x2−y2=32x ^ { 2 } - y ^ { 2 } = 32x2−y2=32

Solve the Problem $x ^ { 2 } - y ^ { 2 } = 32$