For the equation $y ^ { 2 } - 4 x ^ { 2 } = 64$ , find the intercepts and the equations of the asymptotes.
A)intercepts: (2, 0), (-2, 0)asymptotes: y = -8 x , y = 8 x
B)intercepts: (8, 0), (-8, 0)asymptotes: y = -2 x , y = 2 x
C)intercepts: (8, 0), (-8, 0)asymptotes: x = -2 y , x = 2 y
D)intercepts: (0, 8), (0, -8)asymptotes: x = -2 y , x = 2 y
E)intercepts: (0, 8), (0, -8)asymptotes: y = -2 x , y = 2 x

Question

Quizplus · Accepted Answer

To find the intercepts, we can set either $x$ or $y$ equal to 0 and solve for the other variable. Setting $x$ equal to 0, we get $y^2=64$, which gives us $y=\pm8$, so the intercepts are (0,8) and (0,-8).

To find the equations of the asymptotes, we can start by dividing both sides of the equation by $4x^2$ to get $\frac{y^2}{4x^2}-1=1$. Taking the limit as $x$ approaches infinity, we see that $\frac{y}{2x}$ approaches either 1 or -1 depending on the sign of $y$, so the equations of the asymptotes are $y=2x$ and $y=-2x$. Therefore, the best choice is E.

For the Equation y2−4x2=64y ^ { 2 } - 4 x ^ { 2 } = 64y2−4x2=64

For the Equation $y ^ { 2 } - 4 x ^ { 2 } = 64$