Question

graph the ellipse and locate the foci.
\\(\frac{x^2}{\frac{9}{4}} + \frac{y^2}{\frac{25}{4}} = 1\\)

choose the correct graph below.
\\(\bigcirc\\) a.
\\(\bigcirc\\) b.
\\(\bigcirc\\) c.

the foci are located at \\(\square\\).
(type ordered pairs. use a comma to separate answers as needed. simplify your answers. type an exact answer, using radicals as needed)

Explanation:

Step1: Identify Ellipse Standard Form

The given equation is \(\frac{x^2}{\frac{9}{4}}+\frac{y^2}{\frac{25}{4}} = 1\). The standard form of an ellipse is \(\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\) (vertical major axis, \(a > b\)), where \(a^2=\frac{25}{4}\), \(b^2=\frac{9}{4}\). So \(a=\frac{5}{2}\), \(b=\frac{3}{2}\).

Step2: Calculate Distance to Foci (\(c\))

Use the relationship \(c^2=a^2 - b^2\). Substitute \(a^2=\frac{25}{4}\), \(b^2=\frac{9}{4}\):
\(c^2=\frac{25}{4}-\frac{9}{4}=\frac{16}{4}=4\), so \(c = 2\).

Step3: Locate Foci

For a vertical major axis ellipse centered at \((0,0)\), foci are at \((0,\pm c)\). Thus, foci are \((0, 2)\) and \((0, -2)\).

Answer:

\((0, 2)\), \((0, -2)\)