Explanation
We are given the following:
[tex]\begin{gathered} Center(h,k)\to(0,0) \\ Vertices:(-4,0)and(4,0) \\ Foci:(-3,0)and(3,0) \end{gathered}[/tex]
We are required to determine the equation of the ellipse.
We know that the general equation of an ellipse is of the form:
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]
We can determine the equation of the ellipse as follows:
[tex]\begin{gathered} Recall\text{ }that: \\ (h-4)^2=a^2 \\ (0-4)^2=a^2 \\ a^2=16 \\ \\ Also, \\ (h-3)^2=a^2-b^2 \\ (0-3)^2=16-b^2 \\ 9=16-b^2 \\ \therefore b^2=16-9=7 \\ \\ the\text{ }equation\text{ }becomes \\ \frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \\ \frac{(x-0)^2}{16}+\frac{(y-0)^2}{7}=1 \\ \frac{x^2}{16}+\frac{y^2}{7}=1 \end{gathered}[/tex]
Hence, the answer is:
[tex]\frac{x^{2}}{16}+\frac{y^{2}}{7}=1[/tex]
