An ellipse is represented by the equation:
((x - 5)^2)/625 + ((y - 4)^2)/225 = 1
Each directrix of this ellipse is a ________________________ from the center on the major axis.
A) Horizontal line that is 20 units
B) Vertical line that is 20 units
C) Horizontal line that is 31.25 units
D) Vertical line that is 31.25 units?
Answer (2)
The directrices of the given ellipse are vertical lines located 31.25 units from the center. The calculations reveal that each directrix is positioned at that distance due to the properties of the ellipse. Thus, the correct choice is D) Vertical line that is 31.25 units.
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To solve the problem regarding the directrix of the ellipse given by the equation:
625 ( x − 5 ) 2 + 225 ( y − 4 ) 2 = 1
Let's start by identifying the components of this ellipse:
Center : The center of the ellipse is ( h , k ) = ( 5 , 4 ) .
Semi-major and Semi-minor Axes :
The equation compares to a 2 ( x − h ) 2 + b 2 ( y − k ) 2 = 1 . Here, since 225"> 625 > 225 , a 2 = 625 and b 2 = 225 .
Thus, a = 625 = 25 and b = 225 = 15 .
Since b"> a > b , the major axis is horizontal.
Directrices : In a horizontal ellipse, the directrices are vertical lines. The formula to find the directrices is x = h ± c a 2 , where c (distance from the center to the foci) is given by c = a 2 − b 2 .
Calculate c :
c = 625 − 225 = 400 = 20 .
Determine the Directrices :
The directrices are x = h ± c a 2 = 5 ± 20 625 .
Simplifying, 20 625 = 31.25 . Thus, the directrices are x = 5 + 31.25 and x = 5 − 31.25 .
Therefore, each directrix of this ellipse is a vertical line that is 31.25 units from the center on the major axis.
The correct multiple choice answer is therefore (D) Vertical line that is 31.25 units.
