Question

find the standard form of the equation of the ellipse and give the location of its foci. type the standard form of the equation. \\(\frac{x^2}{36}+\frac{y^2}{4}=1\\) (type an equation. simplify your answer.) type the locations of the foci. (type ordered pairs. use a comma to separate answers. type exact answers, using radicals as needed. simplify your answers.)

Explanation:

Step1: Recall ellipse standard form

The standard form of an ellipse centered at the origin with a horizontal major axis is \(\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1\), where \(a>b>0\). The foci are at \((\pm c,0)\), and \(c^2=a^2 - b^2\).

Step2: Identify \(a^2\) and \(b^2\)

From the equation \(\frac{x^2}{36}+\frac{y^2}{4}=1\), we have \(a^2 = 36\) (so \(a = 6\)) and \(b^2=4\) (so \(b = 2\)).

Step3: Calculate \(c\)

Using \(c^2=a^2 - b^2\), substitute \(a^2 = 36\) and \(b^2 = 4\):
\(c^2=36 - 4=32\), so \(c=\sqrt{32}=4\sqrt{2}\).

Step4: Find foci coordinates

Since the major axis is horizontal (centered at the origin), the foci are at \((\pm c,0)\), which is \((\pm4\sqrt{2},0)\).

Answer:

\((-4\sqrt{2},0),(4\sqrt{2},0)\)