Question

graph the ellipse and locate the foci.
3x² = 15 - 5y²

choose the correct graph of the ellipse.
a. graph of an ellipse centered at the origin b. graph of an ellipse centered at the origin

the foci of the ellipse are
(use a comma to separate answers as needed. type ordered pairs. type exact ans

Explanation:

Step1: Rewrite the equation in standard form

First, we start with the given equation \(3x^{2}=15 - 5y^{2}\). We want to rewrite it in the standard form of an ellipse, which is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) (for an ellipse centered at the origin).

Add \(5y^{2}\) to both sides: \(3x^{2}+5y^{2}=15\). Then divide both sides by 15 to get 1 on the right - hand side: \(\frac{3x^{2}}{15}+\frac{5y^{2}}{15}=1\), which simplifies to \(\frac{x^{2}}{5}+\frac{y^{2}}{3}=1\).

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

For the standard form of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) (where \(a>b>0\) and the major axis is along the \(x\) - axis), we have \(a^{2}=5\) and \(b^{2}=3\).

Step3: Calculate \(c\) (distance from center to foci)

The relationship between \(a\), \(b\), and \(c\) for an ellipse is \(c^{2}=a^{2}-b^{2}\). Substitute \(a^{2}=5\) and \(b^{2}=3\) into the formula: \(c^{2}=5 - 3=2\). Then take the square root of both sides: \(c=\sqrt{2}\) (we take the positive square root since \(c\) represents a distance).

Step4: Determine the coordinates of the foci

Since the major axis is along the \(x\) - axis (because \(a^{2}\) is under the \(x^{2}\) term) and the center of the ellipse is at the origin \((0,0)\), the foci are located at \((\pm c,0)\), which is \((\pm\sqrt{2},0)\).

Answer:

The foci of the ellipse are \((\sqrt{2},0),(-\sqrt{2},0)\)