The equation of the ellipse is given by
[tex]\[
\frac{(x-5)^2}{625} + \frac{(y-4)^2}{225} = 1
\][/tex]
From this equation, we can identify several key parameters:
1. The center [tex]\((h, k)\)[/tex] of the ellipse is [tex]\((5, 4)\)[/tex].
2. The semi-major axis length [tex]\( a \)[/tex] and semi-minor axis length [tex]\( b \)[/tex] can be found from the denominators of the ellipse equation.
Given:
[tex]\[
a^2 = 625 \implies a = \sqrt{625} = 25
\][/tex]
[tex]\[
b^2 = 225 \implies b = \sqrt{225} = 15
\][/tex]
Since [tex]\( 625 > 225 \)[/tex], the major axis is along the x-axis.
The focal distance [tex]\( c \)[/tex] is calculated from the relationship between [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
c^2 = a^2 - b^2
\][/tex]
[tex]\[
c^2 = 625 - 225 = 400 \implies c = \sqrt{400} = 20
\][/tex]
The distance from the center of the ellipse to each directrix along the major axis is given by:
[tex]\[
a^2 / c = 625 / 20 = 31.25
\][/tex]
Thus, the directrices are vertical lines at a distance of 31.25 units from the center along the major axis.
Therefore, the correct description is:
Each directrix of this ellipse is a vertical line that is 31.25 units from the center on the major axis.
