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) identify the endpoints of the minor axis.
(f) identify 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 } } { 16 } + \frac { y ^ { 2 } } { 81 } = 1 \\)

Explanation:

(a) Center of the ellipse

The standard form of an ellipse is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (for vertical major axis). For the given equation \(\frac{x^{2}}{16}+\frac{y^{2}}{81}=1\), comparing with \(\frac{(x - 0)^2}{4^2}+\frac{(y - 0)^2}{9^2}=1\), the center \((h,k)=(0,0)\)

(b) Value of \(a\)

Since the equation is \(\frac{x^{2}}{16}+\frac{y^{2}}{81}=1\) and for \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\), we have \(a^{2}=81\), so \(a = 9\)

(c) Value of \(b\)

Since the equation is \(\frac{x^{2}}{16}+\frac{y^{2}}{81}=1\) and for \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\), we have \(b^{2}=16\), so \(b = 4\)

(d) Vertices

For an ellipse \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) with center \((h,k)\), vertices are \((h,k\pm a)\). Here \(h = 0,k = 0,a = 9\), so vertices are \((0,9)\) and \((0,- 9)\)

(e) Endpoints of the minor axis

For an ellipse \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) with center \((h,k)\), endpoints of the minor axis are \((h\pm b,k)\). Here \(h = 0,k = 0,b = 4\), so endpoints are \((4,0)\) and \((-4,0)\)

(f) Foci

We know the relationship \(c^{2}=a^{2}-b^{2}\). Given \(a = 9,b = 4\), then \(c^{2}=81 - 16=65\), so \(c=\sqrt{65}\). For \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\), foci are \((h,k\pm c)\). Here \(h = 0,k = 0,c=\sqrt{65}\), so foci are \((0,\sqrt{65})\) and \((0,-\sqrt{65})\)

(g) Length of the major axis

The length of the major axis is \(2a\). Since \(a = 9\), length of major axis \(=2\times9 = 18\)

(h) Length of the minor axis

The length of the minor axis is \(2b\). Since \(b = 4\), length of minor axis \(=2\times4=8\)

(i) Graphing the ellipse

1. Plot the center at \((0,0)\)
2. Plot the vertices at \((0,9)\) and \((0,-9)\)
3. Plot the endpoints of the minor axis at \((4,0)\) and \((-4,0)\)
4. Sketch the ellipse passing through these points

Answer:

(a) Center: \((0,0)\)
(b) \(a = 9\)
(c) \(b = 4\)
(d) Vertices: \((0,9)\) and \((0,-9)\)
(e) Endpoints of minor axis: \((4,0)\) and \((-4,0)\)
(f) Foci: \((0,\sqrt{65})\) and \((0,-\sqrt{65})\)
(g) Length of major axis: \(18\)
(h) Length of minor axis: \(8\)
(i) Graph: Plot center \((0,0)\), vertices \((0,\pm9)\), minor - axis endpoints \((\pm4,0)\) and sketch the ellipse.