To find the equation of the ellipse, we start with understanding the provided geometric features.
### Step-by-Step Solution:
1. Identify the Center:
- The center of the ellipse is given as [tex]\((h, k) = (2, 2)\)[/tex].
2. Determine Length of the Semi-Major Axis (a):
- The vertices are [tex]\((-3, 2)\)[/tex] and [tex]\((7, 2)\)[/tex].
- The distance between the vertices along the x-axis is [tex]\(7 - (-3) = 10\)[/tex].
- The semi-major axis (a) is half of this distance:
[tex]\[
a = \frac{10}{2} = 5
\][/tex]
3. Determine the Distance Between the Foci:
- The foci are at [tex]\((-1, 2)\)[/tex] and [tex]\((5, 2)\)[/tex].
- The distance between the foci (2c) along the x-axis is [tex]\(5 - (-1) = 6\)[/tex].
- Therefore, the value of [tex]\(c\)[/tex]:
[tex]\[
c = \frac{6}{2} = 3
\][/tex]
4. Calculate the Semi-Minor Axis (b):
- Using the relationship for ellipses [tex]\(a^2 = b^2 + c^2\)[/tex]:
[tex]\[
5^2 = b^2 + 3^2
\][/tex]
[tex]\[
25 = b^2 + 9
\][/tex]
[tex]\[
b^2 = 16
\][/tex]
[tex]\[
b = \sqrt{16} = 4
\][/tex]
5. Form the Standard Equation of the Ellipse:
- The standard form of the equation of an ellipse centered at [tex]\((h, k)\)[/tex] with semi-major axis [tex]\(a\)[/tex] and semi-minor axis [tex]\(b\)[/tex] is:
[tex]\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\][/tex]
- Substituting the values [tex]\(h = 2\)[/tex], [tex]\(k = 2\)[/tex], [tex]\(a = 5\)[/tex], and [tex]\(b = 4\)[/tex]:
[tex]\[
\frac{(x-2)^2}{5^2} + \frac{(y-2)^2}{4^2} = 1
\][/tex]
[tex]\[
\frac{(x-2)^2}{25} + \frac{(y-2)^2}{16} = 1
\][/tex]
Thus, the equation of the ellipse is:
[tex]\[
\left(\frac{(x - 2.0)^2}{25.0}\right) + \left(\frac{(y - 2)^2}{16.0}\right) = 1
\][/tex]
