Question

an equation of an ellipse is given.
(a) identify the center of the ellipse.
(b) determine the value of ( a ).
(c) determine the value of ( b ).
(d) identify the vertices.
(e) identity the endpoints of the minor axis.
(f) identity the foci.
(g) determine the length of the major axis.
(h) determine the length of the minor axis.
(i) graph the ellipse.
express numbers in exact, simplest form.
( \frac{x^{2}}{64}+\frac{y^{2}}{36}=1 )

Explanation:

Step1: Recall the standard form of an ellipse

The standard form of an ellipse is \(\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1\) (for a horizontal major axis) or \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (for a vertical major axis), where \((h,k)\) is the center. For the given equation \(\frac{x^{2}}{64}+\frac{y^{2}}{36}=1\), we can rewrite it as \(\frac{(x - 0)^2}{8^{2}}+\frac{(y - 0)^2}{6^{2}}=1\)

Step2: (a) Find the center

Comparing with the standard form \(\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1\), the center \((h,k)=(0,0)\)

Step3: (b) Find the value of \(a\)

Since \(a^{2}=64\) and \(a>0\), then \(a = 8\)

Step4: (c) Find the value of \(b\)

Since \(b^{2}=36\) and \(b>0\), then \(b = 6\)

Step5: (d) Find the vertices

For an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a>b\)), the vertices are \((\pm a,0)\). Substituting \(a = 8\), the vertices are \((\pm8,0)\)

Step6: (e) Find the endpoints of the minor axis

For an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a>b\)), the endpoints of the minor axis are \((0,\pm b)\). Substituting \(b = 6\), the endpoints are \((0,\pm6)\)

Step7: (f) Find the foci

We know the relationship \(c^{2}=a^{2}-b^{2}\). Substitute \(a = 8\) and \(b = 6\), then \(c^{2}=64 - 36=28\), so \(c=\sqrt{28}=2\sqrt{7}\). The foci are \((\pm c,0)=(\pm2\sqrt{7},0)\)

Step8: (g) Find the length of the major axis

The length of the major axis is \(2a\). Substitute \(a = 8\), so the length is \(2\times8 = 16\)

Step9: (h) Find the length of the minor axis

The length of the minor axis is \(2b\). Substitute \(b = 6\), so the length is \(2\times6=12\)

Step10: (i) Graph the ellipse

• Plot the center at \((0,0)\)
• Plot the vertices at \((\pm8,0)\)
• Plot the endpoints of the minor axis at \((0,\pm6)\)
• Sketch the ellipse passing through these points

Answer:

(a) Center: \((0,0)\)
(b) \(a = 8\)
(c) \(b = 6\)
(d) Vertices: \((8,0)\) and \((- 8,0)\)
(e) Endpoints of the minor axis: \((0,6)\) and \((0,-6)\)
(f) Foci: \((2\sqrt{7},0)\) and \((-2\sqrt{7},0)\)
(g) Length of the major axis: \(16\)
(h) Length of the minor axis: \(12\)
(i) Graph: Plot center \((0,0)\), vertices \((\pm8,0)\), minor - axis endpoints \((0,\pm6)\) and sketch the ellipse.