Find the vertices and asymptotes of the hyperbola. $25 y ^ { 2 } - 4 x ^ { 2 } = 100$
A)vertices: (±2,0), asymptote: $y = \pm \frac { 5 } { 2 } x$
B)vertices: (0,±2), asymptote: $y = \pm \frac { 2 } { 5 } x$
C)vertices: (±2,0), asymptote: $y = \pm \frac { 2 } { 5 } x$
D)vertices: (0,±2), asymptote: $y = \pm \frac { 5 } { 2 } x$
E)vertices: (±2,5), asymptote: $y = \pm \frac { 2 } { 5 } x$

Question

Quizplus · Accepted Answer

The given equation is $25y^2 - 4x^2 = 100$. To find the vertices and asymptotes, we first rewrite the equation in standard form for a hyperbola, which is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where $2a$ is the distance between the vertices along the y-axis, and $2b$ is the distance between the vertices along the x-axis for hyperbolas opening up and down. Comparing, we get $a^2 = 4$ and $b^2 = 25$, so $a = 2$ and $b = 5$. Since $y^2$ is positive, the hyperbola opens up and down, and the vertices are at $(0, \pm a) = (0, \pm 2)$. The asymptotes for a hyperbola in this form are given by $y = \pm \frac{a}{b}x$, which simplifies to $y = \pm \frac{2}{5}x$.

Find the Vertices and Asymptotes of the Hyperbola 25y2−4x2=10025 y ^ { 2 } - 4 x ^ { 2 } = 10025y2−4x2=100

Find the Vertices and Asymptotes of the Hyperbola $25 y ^ { 2 } - 4 x ^ { 2 } = 100$