Let's solve the problem step-by-step to write the equation of the ellipse.
1. Identify the center of the ellipse:
The center [tex]\((h, k)\)[/tex] of the ellipse is given as [tex]\((-2, -3)\)[/tex].
2. Determine the distance from the center to the focus (c):
The focus is located at [tex]\((1, -3)\)[/tex]. Since the focus and the center share the same y-coordinate, the distance [tex]\(c\)[/tex] is simply the difference in their x-coordinates:
[tex]\[
c = |h - \text{focus}_x| = |-2 - 1| = | -3 | = 3
\][/tex]
3. Determine the distance from the center to the vertex (a):
The vertex is at [tex]\((-7, -3)\)[/tex]. Again, as both the vertex and the center share the same y-coordinate, the distance [tex]\(a\)[/tex] is just the difference in the x-coordinates:
[tex]\[
a = |h - \text{vertex}_x| = |-2 - (-7)| = |-2 + 7| = 5
\][/tex]
4. Calculate the distance from the center to the co-vertex (b):
We use the relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for ellipses ([tex]\(c^2 = a^2 - b^2\)[/tex]):
[tex]\[
a^2 = 5^2 = 25
\][/tex]
[tex]\[
c^2 = 3^2 = 9
\][/tex]
[tex]\[
b^2 = a^2 - c^2 = 25 - 9 = 16
\][/tex]
Hence,
[tex]\[
b = \sqrt{b^2} = \sqrt{16} = 4
\][/tex]
5. Form the equation of the ellipse:
The standard form for the equation of an ellipse centered at [tex]\((h, k)\)[/tex] with semi-major axis length [tex]\(a\)[/tex] and semi-minor axis length [tex]\(b\)[/tex] is:
[tex]\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\][/tex]
Substituting [tex]\(h = -2\)[/tex], [tex]\(k = -3\)[/tex], [tex]\(a^2 = 25\)[/tex], and [tex]\(b^2 = 16\)[/tex], the equation becomes:
[tex]\[
\frac{(x + 2)^2}{25} + \frac{(y + 3)^2}{16} = 1
\][/tex]
So, the equation of the ellipse is:
[tex]\[
\boxed{\frac{(x + 2)^2}{25} + \frac{(y + 3)^2}{16} = 1}
\][/tex]
