Question

graph the ellipse and locate the foci.
\\(\frac{x^2}{36} + \frac{y^2}{49} = 1\\)

choose the correct graph below.
a. graph
b. graph
c. graph
d. graph

locate the foci.
(type ordered pairs. use a comma to separate answers. type exact answers, using radicals as needed. simplify your answers.)

Explanation:

Step1: Identify ellipse standard form

The equation is \(\frac{x^2}{36}+\frac{y^2}{49} = 1\), which matches \(\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\) (since \(a^2 > b^2\), major axis is vertical). Here, \(a^2 = 49\), so \(a = 7\); \(b^2 = 36\), so \(b = 6\).

Step2: Calculate distance to foci (\(c\))

Use \(c^2=a^2 - b^2\). Substitute \(a^2 = 49\), \(b^2 = 36\): \(c^2=49 - 36=13\), so \(c=\sqrt{13}\).

Step3: Locate foci

For vertical major axis, foci are at \((0, \pm c)\), so \((0, \sqrt{13})\) and \((0, -\sqrt{13})\).

Step4: Identify correct graph

The ellipse has vertical major axis (taller along y - axis). Among the options, the graph with vertical major axis (e.g., taller in y - direction) is the correct one (visually, check which graph is stretched vertically).

Answer:

The foci are \((0, \sqrt{13})\), \((0, -\sqrt{13})\) and the correct graph is the one with vertical major axis (e.g., the appropriate option among A - D with vertical stretch, likely the one labeled as having vertical elongation).