Question

hyperbolas
now try a similar problem on your own. remember, it is easier to identify (a^2) and (b^2) if the equation of the hyperbola is written in standard form.
what are the foci of the graph (36y^2 - 25x^2 = 900)?
a. ((0, pm sqrt{11}))
b. ((pm sqrt{61}, 0))
c. ((pm sqrt{11}, 0))
d. ((0, pm sqrt{61}))

Explanation:

Step1: Convert to standard form

Divide all terms by 900:
$$\frac{36y^2}{900} - \frac{25x^2}{900} = \frac{900}{900}$$
$$\frac{y^2}{25} - \frac{x^2}{36} = 1$$

Step2: Identify $a^2$ and $b^2$

For vertical hyperbola $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$:
$a^2=25$, $b^2=36$

Step3: Calculate $c$ using $c^2=a^2+b^2$

$$c^2=25+36=61$$
$$c=\sqrt{61}$$

Step4: State foci coordinates

Vertical hyperbola foci: $(0,\pm c)=(0,\pm\sqrt{61})$

Answer:

D. $(0,\pm\sqrt{61})$