An ellipse is represented by the equation:
\[(y - 3)^2 / 169 + (x + 6)^2 / 144 = 1\]
Each directrix of this ellipse is a ________________________ from the center on the major axis.
Options:
(A) horizontal line that is 5 units
(B) vertical line that is 5 units
(C) horizontal line that is 33.8 units
(D) vertical line that is 33.8 units
Answer (2)
The ellipse's center is at (-6, 3) and the distance to the directrix is 33.8 units from the center along the major axis. The directrix is a vertical line because the major axis is vertical. The correct answer is (D) vertical line that is 33.8 units.
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To find the directrices of the ellipse represented by the equation:
169 ( y − 3 ) 2 + 144 ( x + 6 ) 2 = 1
we first need to recognize the components of the ellipse equation in standard form, a 2 ( y − k ) 2 + b 2 ( x − h ) 2 = 1 for a vertically oriented ellipse, where b"> a > b (since 169 > 144), and the center is at ( h , k ) .
Here, the center of the ellipse is at ( − 6 , 3 ) .
The semi-major axis is along the y -axis due to a 2 = 169 (since it's larger than 144), and its length is a = 169 = 13 .
The semi-minor axis is along the x -axis, and its length is b = 144 = 12 .
For ellipses, the distance of the directrix from the center can be found using the formula relevant to its vertical orientation (major axis along the y-axis):
distance of directrix = c a 2 = a 2 − b 2 a 2
First, find the value of c :
c 2 = a 2 − b 2 = 169 − 144 = 25
c = 25 = 5
Now, calculate the distance:
distance = 5 169 = 33.8
Since the major axis is vertical, the directrices are vertical lines. Therefore, each directrix is a vertical line that is 33.8 units from the center on the major axis.
Thus, the correct choice is (D) vertical line that is 33.8 units .
