Sure! Let's start solving the problem step-by-step. The given equation of the ellipse is:
[tex]\[
\frac{(x-5)^2}{625} + \frac{(y-4)^2}{225} = 1.
\][/tex]
### Step 1: Identify the Parameters of the Ellipse
- Center [tex]\((h, k)\)[/tex]: [tex]\((5, 4)\)[/tex]
- Semi-major axis [tex]\(a\)[/tex]: The denominator under the [tex]\(x\)[/tex] term [tex]\(\sqrt{625} = 25\)[/tex]
- Semi-minor axis [tex]\(b\)[/tex]: The denominator under the [tex]\(y\)[/tex] term [tex]\(\sqrt{225} = 15\)[/tex]
### Step 2: Determine Which Axis is the Major Axis
Since [tex]\(625 > 225\)[/tex], the semi-major axis [tex]\(a = 25\)[/tex] lies along the [tex]\(x\)[/tex]-axis. Therefore, the ellipse is oriented horizontally.
### Step 3: Calculate the Focal Distance
The focal distance [tex]\(c\)[/tex] is found using the relationship:
[tex]\[
c = \sqrt{a^2 - b^2}
\][/tex]
where [tex]\(a = 25\)[/tex] and [tex]\(b = 15\)[/tex].
[tex]\[
c = \sqrt{25^2 - 15^2} = \sqrt{625 - 225} = \sqrt{400} = 20
\][/tex]
### Step 4: Determine the Directrix Distance
For horizontal ellipses, the distance of the directrix from the center is calculated using:
[tex]\[
\text{Directrix distance} = \frac{a^2}{c}
\][/tex]
where [tex]\(a = 25\)[/tex] and [tex]\(c = 20\)[/tex].
[tex]\[
\text{Directrix distance} = \frac{25^2}{20} = \frac{625}{20} = 31.25
\][/tex]
### Step 5: Interpret the Result
Since the major axis is horizontal, the directrix lines are vertical and located 31.25 units away from the center on each side along the [tex]\(x\)[/tex]-axis.
So the correct answer is:
[tex]\[
\boxed{\text{vertical line that is 31.25 units}}
\][/tex]
