Question

analyze the equation. that is, find the center, vertices, and foci of the ellipse, and graph it

\\( \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1 \\)

(0,0) (type an ordered pair.)

what are the coordinates of the vertices?

(4,0), (-4,0) (type an ordered pair. type an exact answer for each coordinate, using radicals as needed. use a comma to separate answers as needed.)

what are the coordinates of the foci?

\\( ( 2 \sqrt { 3 } , 0 ) , ( - 2 \sqrt { 3 } , 0 ) \\) (type an ordered pair. type an exact answer for each coordinate, using radicals as needed. use a comma to separate answers as needed.)

choose the correct graph

Explanation:

Step1: Recall the standard form of an ellipse

The standard form of an ellipse is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) (\(a>b>0\)). For the equation \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\), we have \(a^{2}=16\), so \(a = 4\), and \(b^{2}=4\), so \(b = 2\). The center of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \((0,0)\). The vertices of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a>b>0\)) are \((\pm a,0)\). Substituting \(a = 4\), the vertices are \((4,0)\) and \((-4,0)\).

Step2: Calculate the foci

The relationship for the foci of an ellipse is \(c^{2}=a^{2}-b^{2}\). Given \(a^{2}=16\) and \(b^{2}=4\), then \(c^{2}=16 - 4=12\), so \(c=\sqrt{12}=2\sqrt{3}\). The foci of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a>b>0\)) are \((\pm c,0)\). Substituting \(c = 2\sqrt{3}\), the foci are \((2\sqrt{3},0)\) and \((-2\sqrt{3},0)\).

Step3: Analyze the graph

For the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\) (\(a = 4\), \(b = 2\)), it is a horizontal ellipse (since \(a^{2}\) is under the \(x^{2}\) term). The length of the major - axis is \(2a=8\) (along the \(x\) - axis) and the length of the minor - axis is \(2b = 4\) (along the \(y\) - axis).

Answer:

• Vertices: \((4,0),(-4,0)\)
• Foci: \((2\sqrt{3},0),(-2\sqrt{3},0)\)
• Graph: Option B (since it is a horizontal ellipse with center at \((0,0)\), vertices at \((\pm4,0)\) and co - vertices at \((0,\pm2)\))