Question

use vertices and asymptotes to graph the hyperbola. locate the foci and find the equations of the asymptotes. 36y² - 4x² = 144 graph the hyperbola. choose the correct graph below. a. graph, b. graph, c. graph the foci is/are at the point(s) (type an ordered pair. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Rewrite in standard form

Divide equation by 144: $\frac{y^2}{4} - \frac{x^2}{36} = 1$
This is vertical hyperbola: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
$a^2=4\Rightarrow a=2$, $b^2=36\Rightarrow b=6$

Step2: Find foci coordinates

Foci at $(0,\pm c)$ where $c^2=a^2+b^2$
$c^2=4+36=40\Rightarrow c=2\sqrt{10}$
Foci: $(0,2\sqrt{10}), (0,-2\sqrt{10})$

Step3: Identify correct graph

Vertical hyperbola opens up/down (matches graph B)

Answer:

Correct graph: B
Foci: $(0, 2\sqrt{10}), (0, -2\sqrt{10})$
Asymptotes: $y=\pm\frac{1}{3}x$