The foci are on the [latex]x[/latex]-axis, so the major axis is the [latex]x[/latex]-axis. Thus the equation will have the form:

[latex]\dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1[/latex]

The vertices are [latex](\pm 8,0)[/latex], so [latex]a=8[/latex] and [latex]a^2=64[/latex].

The foci are [latex](\pm 5,0)[/latex], so [latex]c=5[/latex] and [latex]c^2=25[/latex].

We know that the vertices and foci are related by the equation [latex]c^2=a^2-b^2[/latex]. Solving for [latex]b^2[/latex] we have

[latex]\begin{align}&c^2=a^2-b^2&& \\ &25 = 64 - b^2 && \text{Substitute for }c^2 \text{ and }a^2. \\ &b^2=39 && \text{Solve for } b^2. \end{align}[/latex]

Now we need only substitute [latex]a^2 = 64[/latex] and [latex]b^2=39[/latex] into the standard form of the equation. The equation of the ellipse is

[latex]\dfrac{{x}^{2}}{64}+\dfrac{{y}^{2}}{39}=1[/latex]

First, we determine the position of the major axis. Because [latex]25>9[/latex], the major axis is on the [latex]y[/latex]-axis. It is vertically oriented.Comparing the equation with the standard form:

[latex]a^2 = 25 \text{ and }b^2 = 9[/latex]

Therefore: [latex]a = \sqrt{25} = 5[/latex] and [latex]b = \sqrt{9} = 3[/latex].

First, let’s find the value of [latex]c[/latex] to find the coordinates of the foci:

[latex]\begin{align}c&=\pm \sqrt{{a}^{2}-{b}^{2}} \\ &=\pm \sqrt{25 - 9} \\ &=\pm \sqrt{16} \\ &=\pm 4 \end{align}[/latex]

Key Components:

[latex]\begin{align*} &\text{Center: } (0, 0) \\ &\text{Vertices: } (0, \pm 5) \\ &\text{Co-vertices: } (\pm 3, 0) \\ &\text{Foci: } (0, \pm 4) \\ \end{align*}[/latex]

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.

First, use algebra to rewrite the equation in standard form.

[latex]\begin{gathered} 4{x}^{2}+25{y}^{2}=100 \\[1.5mm] \dfrac{4{x}^{2}}{100}+\dfrac{25{y}^{2}}{100}=\dfrac{100}{100} \\[1.5mm] \dfrac{{x}^{2}}{25}+\dfrac{{y}^{2}}{4}=1 \end{gathered}[/latex]

Next, we determine the position of the major axis. Because [latex]25>4[/latex], the major axis is on the [latex]x[/latex]-axis. Therefore, the equation is in the form [latex]\dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1[/latex], where [latex]{a}^{2}=25[/latex] and [latex]{b}^{2}=4[/latex]. It follows that:

• the center of the ellipse is [latex]\left(0,0\right)[/latex]
• the coordinates of the vertices are [latex]\left(\pm a,0\right)=\left(\pm \sqrt{25},0\right)=\left(\pm 5,0\right)[/latex]
• the coordinates of the co-vertices are [latex]\left(0,\pm b\right)=\left(0,\pm \sqrt{4}\right)=\left(0,\pm 2\right)[/latex]
• the coordinates of the foci are [latex]\left(\pm c,0\right)[/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[/latex]. Solving for [latex]c[/latex], we have:

[latex]\begin{align}c&=\pm \sqrt{{a}^{2}-{b}^{2}} \\ &=\pm \sqrt{25 - 4} \\ &=\pm \sqrt{21} \end{align}[/latex]

Therefore the coordinates of the foci are [latex]\left(\pm \sqrt{21},0\right)[/latex].

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.