To find the equation of the ellipse, we use the given information about its vertices and a point lying on it.
1. Determine the length of the major axis and the value of [tex]\(a\)[/tex]:
- The vertices of the ellipse are given as [tex]\((-5,0)\)[/tex] and [tex]\((5,0)\)[/tex].
- The distance between the vertices is [tex]\(10\)[/tex], which is the length of the major axis.
- Since the length of the major axis [tex]\(2a = 10\)[/tex], then [tex]\(a = 5\)[/tex].
2. Determine the value of [tex]\(b\)[/tex]:
- A point on the ellipse is given as [tex]\((0,2)\)[/tex], which means when [tex]\(x=0\)[/tex], [tex]\(y\)[/tex] reaches its maximum value, making [tex]\(y = b\)[/tex].
- Therefore, [tex]\(b = 2\)[/tex].
3. Form the equation of the ellipse:
- The standard form of the equation of an ellipse centered at the origin is given by:
[tex]\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\][/tex]
- Substituting [tex]\(a = 5\)[/tex] and [tex]\(b = 2\)[/tex]:
[tex]\[
\frac{x^2}{5^2} + \frac{y^2}{2^2} = 1
\][/tex]
- Simplifying:
[tex]\[
\frac{x^2}{25} + \frac{y^2}{4} = 1
\][/tex]
4. Identify the correct choice:
- The correct choice is:
[tex]\[
\frac{x^2}{25} + \frac{y^2}{4} = 1
\][/tex]
Thus, the correct answer is [tex]\(\boxed{\frac{x^2}{5^2} + \frac{y^2}{2^2} = 1}\)[/tex], which corresponds to option D.
