To find the standard form of the equation of the ellipse with the given characteristics, let's proceed step by step:
1. Identify the Center of the Ellipse:
The center of the ellipse is given as [tex]\((1, 3)\)[/tex]. In the standard form of the equation of an ellipse [tex]\(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\)[/tex], [tex]\((h, k)\)[/tex] is the center. So, we have:
[tex]\[
h = 1, \quad k = 3
\][/tex]
2. Find the Semi-Major Axis Length ([tex]\(a\)[/tex]):
The vertex given is [tex]\((-6, 3)\)[/tex], which lies on the major axis of the ellipse. The distance from the center to a vertex along the major axis is the length of the semi-major axis, denoted as [tex]\(a\)[/tex]. We can calculate this distance as follows:
[tex]\[
a = |x_{\text{vertex}} - x_{\text{center}}| = |(-6) - 1| = 7
\][/tex]
3. Find the Semi-Minor Axis Length ([tex]\(b\)[/tex]):
The total length of the minor axis is given as 10, so the semi-minor axis length [tex]\(b\)[/tex] is half of this value:
[tex]\[
b = \frac{\text{Length of minor axis}}{2} = \frac{10}{2} = 5
\][/tex]
4. Square the Semi-Major and Semi-Minor Axis Lengths:
To write the standard form of the ellipse equation, we need [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
[tex]\[
a^2 = 7^2 = 49
\][/tex]
[tex]\[
b^2 = 5^2 = 25
\][/tex]
5. Write the Standard Form of the Ellipse Equation:
We now substitute [tex]\(h\)[/tex], [tex]\(k\)[/tex], [tex]\(a^2\)[/tex], and [tex]\(b^2\)[/tex] into the standard form equation:
[tex]\[
\frac{(x - 1)^2}{49} + \frac{(y - 3)^2}{25} = 1
\][/tex]
Therefore, the standard form of the equation of the ellipse is:
[tex]\[
\frac{(x - 1)^2}{49} + \frac{(y - 3)^2}{25} = 1
\][/tex]
