To find the equation of the ellipse with the given properties (Foci at [tex]\((0, \pm 9)\)[/tex] and vertices at [tex]\(( \pm 7,0)\)[/tex]), we will follow these steps:
1. Identify the center, foci, and vertices:
- The center of the ellipse is at the origin [tex]\((0,0)\)[/tex] because the foci and vertices are symmetric around this point.
- The distance from the center to each focus (denoted as [tex]\(c\)[/tex]) is [tex]\(9\)[/tex].
- The distance from the center to each vertex (denoted as [tex]\(a\)[/tex]) is [tex]\(7\)[/tex].
2. Relate the lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- For an ellipse, the relationship between the semi-major axis [tex]\(a\)[/tex], the semi-minor axis [tex]\(b\)[/tex], and the distance to the foci [tex]\(c\)[/tex] is given by the equation:
[tex]\[
c^2 = a^2 - b^2
\][/tex]
Here, we know [tex]\(c\)[/tex] and [tex]\(a\)[/tex], so we can solve for [tex]\(b\)[/tex].
3. Substitute the known values and solve for [tex]\(b\)[/tex]:
- Given [tex]\(c = 9\)[/tex] and [tex]\(a = 7\)[/tex], we calculate:
[tex]\[
c^2 = 9^2 = 81
\][/tex]
[tex]\[
a^2 = 7^2 = 49
\][/tex]
- Using the relationship:
[tex]\[
81 = 49 - b^2
\][/tex]
- Rearrange to solve for [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = 49 - 81
\][/tex]
[tex]\[
b^2 = -32
\][/tex]
- So we get:
[tex]\[
b = \sqrt{-32} = \sqrt{32} \, i = 4\sqrt{2} \, i
\][/tex]
Note: [tex]\(i\)[/tex] is the imaginary unit, indicating that [tex]\(b\)[/tex] is purely imaginary.
4. Write the equation of the ellipse:
- The standard form of the equation of an ellipse centered at the origin is:
[tex]\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\][/tex]
- Substituting [tex]\(a^2 = 49\)[/tex] and [tex]\(b^2 = -32\)[/tex], we get:
[tex]\[
\frac{x^2}{49} + \frac{y^2}{-32} = 1
\][/tex]
Therefore, the equation of the ellipse is:
[tex]\[
\boxed{\frac{x^2}{49} + \frac{y^2}{-32} = 1}
\][/tex]
