Question

use vertices and asymptotes to graph the hyperbola. locate the foci and find the equations of the asymptotes.
\\(\frac{y^2}{4} - \frac{x^2}{49} = 1\\)
graph the hyperbola. choose the correct graph below.
\\(\bigcirc\\) a. graph of hyperbola opening up and down
\\(\bigcirc\\) b. graph of hyperbola opening left and right (selected)
\\(\bigcirc\\) c. graph of hyperbola opening up and down
the foci is/are at the point(s) \\(\square\\).
(type an ordered pair. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Identify hyperbola type and parameters

The equation is $\frac{y^2}{4} - \frac{x^2}{49} = 1$, which is a vertical hyperbola ($\frac{y^2}{a^2} - \frac{x^2}{b^2}=1$). Here, $a^2=4\Rightarrow a=2$, $b^2=49\Rightarrow b=7$.

Step2: Calculate c for foci

Use $c^2=a^2+b^2$: $c^2=4+49=53\Rightarrow c=\sqrt{53}$. Foci are $(0,\pm c)$.

Step3: Determine asymptote equations

For vertical hyperbola: $y=\pm\frac{a}{b}x=\pm\frac{2}{7}x$.

Step4: Match graph

Vertical hyperbola opens up/down, vertices at $(0,\pm2)$. Correct graph is B.

Answer:

B. [Graph B], Foci: $(0,\sqrt{53}), (0,-\sqrt{53})$, Asymptotes: $y=\frac{2}{7}x, y=-\frac{2}{7}x$